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Notes: 1- We ask you kindly to go through the below listed topics of physics and to concentrate on the main concepts and outlines i.e. you don’t need to learn by heart all texts, figures and tables, it is just to give you a sufficient explanation for the different topics and to deepen your understanding to the material. 2- Please notify, for the interview if you choose physics instead of chemistry then you need to study all the below listed topics and concentrate on at least ten topics. 3- Applicant has to choose either (biology and chemistry) or (biology and physics) for the oral part of the entrance examination. TOPICS ON PHYSICS FOR ENTRANCE EXAMINATION 1. Describing motion (distance, speed and acceleration) 2. Newton’s laws of motion 3. Scalar and vector quantities 4. Mechanical work, kinetic and potential energy 5. Elastic and inelastic collisions, conservation of linear momentum 6. Uniform circular motion, centripetal force 7. Mechanical advantage, simple mechanical tools (the inclined plane, the screw, the pulley) 8. Pressure in fluids (Pascal’s principle), Archimedes’ principle, The hydraulic press 9. Harmonic motion, Hooke’s law 10. Wave motion, longitudinal and transverse waves, resonance 11. Kinetic theory of gases, the temperature 12. The first and second laws of thermodynamics 13. The electric field, Coulomb’s law 14. The electric current, Ohm’s law 15. Simple electric circuits. Kirchhoff’s laws 16. Electromagnetic induction, the transformer 17. Propagation of light, reflection and refraction, optical lenses mirrors 18. Radioactivity Textbook: Modern technical Physics by Arthur Beiser th Publisher: Benjamin-Cummings Publishing Company 4 edition (January 1983) ASIN: 080530682X Answers Scalars and Vectors Physics is a mathematical science - that is, the underlying concepts and principles have a mathematical basis. Throughout this tutorial, you will encounter a variety of concepts which have a mathematical basis associated with them. While the emphasis will often be upon the conceptual nature of physics, there will also be considerable and persistent attention given to its mathematical aspect. These two categories can be distinguished from one another by their distinct definitions: Scalars are quantities which are fully described by a magnitude alone, example time.length, Dudtance, speed, temperature, pressure, heat… Vectors are quantities which are fully described by both a magnitude and a direction, example Force, displacement, velocity, acceleration…. Distance and Displacement Distance and displacement are two quantities which may seem to mean the same thing, yet they have distinctly different meanings and definitions. Distance is a scalar quantity which refers to "how much ground an object has covered" during its motion. Displacement is a vector quantity which refers to "how far out of place an object is"; it is the object's change in position. Speed and Velocity Just as distance and displacement have distinctly different meanings (despite their similarities), so do speed and velocity. Speed is a scalar quantity which refers to "how fast an object is moving." A fast-moving object has a high speed while a slow-moving object has a low speed. An object with no movement at all has a zero speed. Velocity is a vector quantity which refers to "the rate at which an object changes its position." Imagine a person moving rapidly - one step forward and one step back - always returning to the original starting position. While this might result in a frenzy of activity, it would also result in a zero velocity. Because the person always returns to the original position, the motion would never result in a change in position. Since velocity is defined as the rate at which the position changes, this motion results in zero velocity. If a person in motion wishes to maximize his/her velocity, then that person must make every effort to maximize the amount that he/she is displaced from his/her original position. Every step must go into moving that person further from where he/she started. For certain, the person should never change directions and begin to return to where he/she started. Describing Speed and Velocity Velocity is a vector quantity. As such, velocity is "direction-aware." When evaluating the velocity of an object, you must keep track of its direction. It would not be enough to say that an object has a velocity of 55 mi/hr. You must include direction information in order to fully describe the velocity of the object. For instance, you must describe an object's velocity as being 55 mi/hr, east. This is one of the essential differences between speed and velocity. Speed is a scalar and does not keep track of direction; velocity is a vector and is direction-aware. The task of describing the direction of the velocity vector is easy! The direction of the velocity vector is the same as the direction in which an object is moving. It does not matter whether the object is speeding up or slowing down, if the object is moving rightwards, then its velocity is described as being rightwards. If an object is moving downwards, then its velocity is described as being downwards. Thus an airplane moving towards the west with a speed of 300 mi/hr has a velocity of 300 mi/hr, west. Note that speed has no direction (it is a scalar) and that velocity is simply the speed with a direction. Average Speed and Average Velocity As an object moves, it often undergoes changes in speed. For example, during an average trip to school, there are many changes in speed. Rather than the speedometer maintaining a steady reading, the needle constantly moves up and down to reflect the stopping and starting and the accelerating and decelerating. At one instant, the car may be moving at 50 mi/hr and at another instant, it may be stopped (i.e., 0 mi/hr). Yet during the course of the trip to school the person might average a speed of 25 mi/hr. The average speed during the course of a motion is often computed using the following equation: Meanwhile, the average velocity is often computed using the equation: Instantaneous Speed Since a moving object often changes its speed during its motion, it is common to distinguish between the average speed and the instantaneous speed. The distinction is as follows: Instantaneous Speed - speed at any given instant in time. Average Speed - average of all instantaneous speeds; found simply by a distance/time ratio. You might think of the instantaneous speed as the speed which the speedometer reads at any given instant in time and the average speed as the average of all the speedometer readings during the course of the trip. Constant Speed Moving objects don't always travel with erratic and changing speeds. Occasionally, an object will move at a steady rate with a constant speed. That is, the object will cover the same distance every regular interval of time. For instance, a cross-country runner might be running with a constant speed of 6 m/s in a straight line. If her speed is constant, then the distance traveled every second is the same. The runner would cover a distance of 6 meters every second. If you measured her position (distance from an arbitrary starting point) each second, you would notice that her position was changing by 6 meters each second. This would be in stark contrast to an object which is changing its speed. An object with a changing speed would be moving a different distance each second. The data tables below depict objects with constant and changing speeds. Acceleration The final mathematical quantity discussed in Lesson 1 is acceleration. An often misunderstood quantity, acceleration has a meaning much different from the meaning sports announcers and other individuals associate with it. The definition of acceleration is: Acceleration is a vector quantity which is defined as "the rate at which an object changes its velocity." An object is accelerating if it is changing its velocity. Sports announcers will occasionally say that a person is accelerating if he/she is moving fast. Yet acceleration has nothing to do with going fast. A person can be moving very fast, and still not be accelerating. Acceleration has to do with changing how fast an object is moving. If an object is not changing its velocity, then the object is not accelerating. The data at the right is representative of an accelerating object – the velocity is changing with respect to time. In fact, the velocity is changing by a constant amount - 10 m/s - in each second of time. Whenever an object's velocity is changing, that object is said to be accelerating; that object has an acceleration. Constant Acceleration Sometimes an accelerating object will change its velocity by the same amount each second. As mentioned before, the data above shows an object changing its velocity by 10 m/s in each consecutive second. This is known as a constant acceleration since the velocity is changing by the same amount each second. An object with a constant acceleration should not be confused with an object with a constant velocity. Don't be fooled! If an object is changing its velocity – whether by a constant amount or a varying amount – it is an accelerating object. An object with a constant velocity is not accelerating. The data tables below depict motions of objects with a constant acceleration and with a changing acceleration. Note that each object has a changing velocity. Since accelerating objects are constantly changing their velocity, you can say that the distance traveled divided by the time taken to travel that distance is not a constant value. A falling object for instance usually accelerates as it falls. If you were to observe the motion of a free-falling object (free fall motion will be discussed in detail later), you would notice that the object averages a velocity of 5 m/s in the first second, 15 m/s in the second second, 25 m/s in the third second, 35 m/s in the fourth second, etc. Our free-falling object would be accelerating at a constant rate. Calculating Acceleration The acceleration of any object is calculated using the equation: This equation can be used to calculate the acceleration of the object whose motion is depicted by the velocity-time data table above. The velocity-time data in the table shows that the object has an acceleration of 10 m/s/s. The calculation is shown below: Acceleration values are expressed in units of velocity per time. Typical acceleration units include the following: m/s/s mi/hr/s Km/hr/s Initially, these units are a little awkward to the newcomer to physics. Yet, they are very reasonable units when you consider the definition of and equation for acceleration. The reason for the units becomes obvious upon examination of the acceleration equation. Since acceleration is a velocity change over a time interval, the units for acceleration are velocity units divided by time units – thus (m/s)/s or (mi/hr)/s. Direction of the Acceleration Vector Acceleration is a vector quantity so it will always have a direction associated with it. The direction of the acceleration vector depends on two factors: whether the object is speeding up or slowing down whether the object is moving in the positive (+) or negative (–) direction The general RULE OF THUMB is: If an object is slowing down, then its acceleration is in the opposite direction of its motion. This RULE OF THUMB can be applied to determine whether the sign of the acceleration of an object is positive or negative, right or left, up or down, etc. Consider the two data tables below. In Example A, the object is moving in the positive direction (i.e., has a positive velocity) and is speeding up. When an object is speeding up, the acceleration is in the same direction as the velocity. Thus, this object has a positive acceleration. In Example B, the object is moving in the negative direction (i.e., has a negative velocity) and is slowing down. When an object is slowing down, the acceleration is in the opposite direction as the velocity. Thus, this object also has a positive acceleration. The Acceleration of Gravity You learned in that a free-falling object is an object which is falling under the sole influence of gravity; such an object has an acceleration on Earth of 9.8 m/s/s, downward. This numerical value for the acceleration of a free-falling object is such an important value that it has been given a special name. It is known as the acceleration of gravity – the acceleration for any object moving under the sole influence of gravity. As a matter of fact, this quantity known as the acceleration of gravity is such an important quantity that physicists have a special symbol to denote it – the symbol g. The numerical value for the acceleration of gravity is most accurately known as 9.8 m/s/s. There are slight variations in this numerical value (to the second decimal place) which are dependent primarily upon altitude. The Physics Classroom will use the approximated value of 10 m/s/s in order to reduce the complexity of the many mathematical tasks performed with this number. By so doing, you will be able to better focus on the conceptual nature of physics without sacrificing too much in the way of numerical accuracy. When the moment arises that you need to be accurate (such as in lab work), use the more accurate value of 9.8 m/s/s. g = 10 m/s/s, downward Recall that acceleration is the rate at which an object changes its velocity. Between any two points in an object's path, acceleration is the ratio of velocity change to the time taken to make that change. To accelerate at 10 m/s/s means to change your velocity by 10 m/s each second. If the velocity and time for a free-falling object being dropped from a position of rest were tabulated, you would notice the following pattern: Time (s) Velocity (m/s)** 0 0 1 10 2 20 3 30 4 40 5 50 (**velocity values are based on using the approximated value of 10 m/s/s for g) The velocity-time data above reveals that the object's velocity is changing by 10 m/s each consecutive second. That is, the free-falling object has an acceleration of 10 m/s/s. Newton's First Law Isaac Newton (a 17th century scientist) put forth three laws which explain why objects move (or don't move) as they do and these three laws have become known as Newton's three laws of motion. The focus of Newton's first law of motion – sometimes referred to as the "law of inertia." Newton's first law of motion is often stated as: An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force. There are two parts to this statement – one which predicts the behavior of stationary objects and the other which predicts the behavior of moving objects. These two parts are summarized in the following diagram. The behavior of all objects can be described by saying that objects tend to "keep on doing what they're doing" (unless acted upon by an unbalanced force). If at rest, they will continue in this same state of rest. If in motion with an eastward velocity of 5 m/s, they will continue in this same state of motion (5 m/s, East). If in motion with a leftward velocity of 2 m/s, they will continue in this same state of motion (2 m/s, left). The state of motion of an object is maintained as long as the object is not acted upon by an unbalanced force. All objects resist changes in their state of motion – they tend to "keep on doing what they're doing." There are many applications of Newton's first law of motion. Consider some of your experiences in an automobile. Have you ever observed the behavior of coffee in a coffee cup filled to the rim while starting a car from rest or while bringing a car to rest from a state of motion? Coffee tends to "keep on doing what it is doing." When you accelerate a car from rest, the road provides an unbalanced force on the spinning wheels to push the car forward; yet the coffee (which is at rest) wants to stay at rest. While the car accelerates forward, the coffee remains in the same position; subsequently, the car accelerates out from under the coffee and the coffee spills in your lap. On the other hand, when braking from a state of motion the coffee continues to move forward with the same speed and in the same direction, ultimately hitting the windshield or the dashboard. Coffee in motion tends to stay in motion. Have you ever experienced inertia (resisting changes in your state of motion) in an automobile while it is braking to a stop? The force of the road on the locked wheels provides the unbalanced force to change the car's state of motion, yet there is no unbalanced force to change your own state of motion. Thus, you continue in motion, sliding forward along the seat. A person in motion tends to stay in motion with the same speed and in the same direction ... unless acted upon by the unbalanced force of a seat belt. Yes, seat belts are used to provide safety for passengers whose motion is governed by Newton's laws. The seat belt provides the unbalanced force which brings you from a state of motion to a state of rest. Perhaps you could speculate what would occur when no seat belt is used. Inertia and Mass Newton's first law of motion states that "An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force," that is, objects "tend to keep on doing what they're doing." In fact, it is the natural tendency of objects to resist changes in their state of motion. This tendency to resist changes in their state of motion is described as inertia. Inertia is the resistance an object to a change in its state of motion. Isaac Newton built on Galileo's thoughts about motion. Newton's first law of motion declares that a force is not needed to keep an object in motion. Slide a book across a table and watch it slide to a stop. The book in motion on the table top does not come to rest because of the absence of a force; rather it is the presence of a force – the force of friction – which brings the book to a halt. In the absence of a frictional force, the book would continue in motion with the same speed and in the same direction – forever! (Or at least to the end of the table top.) A force is not required to keep a moving book in motion; in actuality, it is a force which brings the book to rest. All objects resist changes in their state of motion. All objects have this tendency – they have inertia. But do some objects have more of a tendency to resist changes than others? Yes, absolutely! The tendency of an object to resist changes in its state of motion is dependent upon its mass. Inertia is a quantity which is solely dependent upon mass. The more mass an object has, the more inertia it has – the more tendency it has to resist changes in its state of motion. Suppose that there are two seemingly identical bricks at rest on a table. However, one brick consists of mortar and the other brick consists of Styrofoam. Without lifting the bricks, how could you tell which brick was the Styrofoam brick? You could give the bricks an identical push in an effort to change their state of motion. The brick which offers less resistance is the brick with less inertia – and therefore the brick with less mass (i.e., the Styrofoam brick). A common physics demonstration relies on this principle that the more massive the object, the more it tends to resist changes in its state of motion. The demonstration goes as follows: several massive books are placed upon the physics teacher's head. A wooden board is placed on top of the books and a hammer is used to drive a nail into the board. Due to the large mass of the books, the force of the hammer is sufficiently resisted (inertia). This is demonstrated by the fact that the blow of the hammer is not felt by the teacher. A common variation of this demonstration involves smashing a brick over the teacher's hand using a swift blow of the hammer. The massive brick resists the force and the hand is not hurt at all. (CAUTION: Do not try these demonstrations at home!) Newton's First Law of Motion Balanced and Unbalanced Forces An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force. But what exactly does the phrase "unbalanced force" mean? What is an unbalanced force? In pursuit of an answer, consider a physics book at rest on a table top. There are two forces acting upon the book. One force – the Earth's gravitational pull – exerts a downward force. The second force – the push of the table on the book (sometimes referred to as a normal force) – pushes upward on the book. Since these two forces are of equal magnitude and in opposite directions, they balance each other. The book is said to be at equilibrium. There is no unbalanced force acting upon the book and thus the book maintains its state of motion. When all the forces acting upon an object balance each other, the object will be at equilibrium; it will not accelerate. (Note: diagrams such as the one above are known as free-body diagrams . Consider another example of a balanced force – a person standing upon the ground. There are two forces acting upon the person. The force of gravity exerts a downward force. The push of the floor exerts an upward force. Since these two forces are of equal magnitude and in opposite directions, they balance each other. The person is at equilibrium. There is no unbalanced force acting upon the person and thus the person maintains his/her state of motion. (Note: diagrams such as the one above are known as free-body diagrams.) Now consider a book sliding from left to right across a table top. Sometime in the history of the book, it may have been given a shove and set in motion from its rest position. Or perhaps it acquired its motion by sliding down an incline from an elevated position. Whatever the case, the focus is not upon the history of the book but rather upon the current situation of the book sliding across a table top. The book is in motion and at the moment there is no one pushing it to the right. (Remember: a force is not needed to keep a moving object moving.) The forces acting upon the book are shown below. The force of gravity pulling downwards and the force of the table pushing upwards on the book are of equal magnitude and in opposite directions. These two forces balance each other. However, there is no force present to balance the force of friction. As the book moves to the right, friction acts to the left to slow the book down. This is an unbalanced force; and as such, the book changes its state of motion. The book is not at equilibrium and it subsequently accelerates. Unbalanced forces cause accelerations. In this case, since the unbalanced force is directed opposite to the object's motion, it will cause a deceleration (a slowing down) of the object. To determine if the forces acting upon an object are balanced or unbalanced, an analysis must first be conducted to determine which forces are acting upon the object and in what direction. If two individual forces acting on an object are of equal magnitude and opposite direction, then these forces are said to be balanced. An object is said to be "acted upon by an unbalanced force" only when there is an individual force acting on the object which is not balanced by another force of equal magnitude and in the opposite direction The Meaning of Force A force is a push or pull upon an object resulting from the object's interaction with another object. Whenever there is an interaction between two objects, there is a force acting on each of the objects. When the interaction ceases, the two objects no longer experience a force. Forces only exist as a result of an interaction. For simplicity sake, all forces (interactions) between objects can be placed into two broad categories: 1- contact forces, and 2- forces resulting from action-at-a-distance Contact forces are types of forces in which the two interacting objects are physically in contact with each other. Examples of contact forces include frictional forces, tensional forces, normal forces, air resistance forces, and applied forces. Action-at-a-distance forces are types of forces in which the two interacting objects are not in physical contact with each other, but are able to exert a push or pull despite the physical separation. Examples of action-at-a-distance forces include gravitational forces (e.g., the sun and planets exert a gravitational pull on each other despite their large spatial separation; even when your feet leave the earth and you are no longer in contact with the earth, there is a gravitational pull between you and the Earth), electric forces (e.g., the protons in the nucleus of an atom and the electrons outside the nucleus experience an electrical pull towards each other despite their small spatial separation), and magnetic forces (e.g., two magnets can exert a magnetic pull on each other even when separated by a distance of a few centimeters). Contact Forces Action-at-a-Distance Forces Frictional Force Gravitational Force Tensional Force Electrical Force Normal Force Magnetic Force Air Resistance Force Applied Force Spring Force Force is a quantity which is measured using a standard metric unit known as the Newton. One Newton is the amount of force required to give a 1-kg mass an acceleration of 1 m/s2. A Newton is abbreviated by an "N." If you say "10.0 N," you mean 10.0 Newtons of force. Thus, the following unit equivalency can be stated: Force is a vector quantity. As you learned earlier, a vector quantity is a quantity which has both magnitude and direction. To fully describe the force acting upon an object, you must describe both its magnitude (size) and its direction. Thus, 10 Newtons is not a full description of the force acting upon an object. In contrast, 10 Newtons, downwards is a complete description of the force acting upon an object; both the magnitude (10 Newtons) and the direction (downwards) are given. Because force is a vector and has direction, it is common to represent forces using diagrams in which the force is represented by an arrow. Such vector diagrams were introduced earlier and will be used throughout your study of physics. The size of the arrow is reflective of the magnitude of the force and the direction of the arrow reveals the direction in which the force is acting. (Furthermore, because forces are vectors, the influence of one individual force upon an object is often canceled by the influence of another force acting on the same object. For example, the influence of a 20-Newton upward force acting upon a book is canceled by the influence of a 20-Newton downward force acting upon the book. In such instances, the two individual forces are said to "balance each other"; there would be no unbalanced force acting upon this book. Other situations could be imagined in which two of the individual vector forces cancel each other ("balance"), and a third individual force exists that is not balanced by another force. For example, imagine a book sliding across the rough surface of a table from left to right. The downward force of gravity and the upward force of the table supporting the book are of equal magnitude, act in opposite directions and thus balance each other. However, the force of friction acts leftwards, and there is no rightward force to balance it. In this case, an unbalanced force acts upon the book to change its state of motion and the book slows down. Forces may be placed into two broad categories, based on whether the force resulted from the contact or non-contact of the two interacting objects. Contact Forces Action-at-a-Distance Forces Frictional Force Gravitational Force Tensional Force Electrical Force Normal Force Magnetic Force Air Resistance Force Applied Force Spring Force These types of forces will now be discussed in detail. To read about each force listed above, continue scrolling through this page, or click on its name from the list below. Applied Force Gravitational Force Normal Force Frictional Force Air Resistance Force Tensional Force Spring Force Type of Force Description of Force and its Symbol An applied force is a force which is applied to an object by another object or by a person. If a person is pushing a desk across the room, then there is an applied Applied Force force acting upon the desk. The applied force is the force exerted on the desk by Fapp the person. The force of gravity is the force with which the earth, moon, or other massive body attracts an object towards itself. By definition, this is the weight of the object. All objects upon earth experience a force of gravity which is directed "downward" towards the center of the earth. The force of gravity on an object Gravity Force on earth is always equal to the weight of the object as given by the equation: (also known as Fgrav = m * g Weight) where: Fgrav g = acceleration of gravity = 9.8 m/s2 (on Earth) m = mass (in kg) (Caution: do not confuse weight with mass.) The normal force is the support force exerted upon an object which is in contact with another stable object. For example, if a book is resting upon a surface, then Normal Force the surface is exerting an upward force upon the book in order to support the Fnorm weight of the book. On occasion, a normal force is exerted horizontally between two objects which are in contact with each other. The friction force is the force exerted by a surface as an object moves across it or makes an effort to move across it. The friction force opposes the motion of the object. For example, if a book moves across the surface of a desk, the desk exerts a friction force in the direction opposite to the motion of the book. Friction results when two surfaces are pressed together closely, causing Friction Force attractive intermolecular forces between the molecules of the two different Ffrict surfaces. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together. The friction force can be calculated using the equation: Air resistance is a special type of frictional force which acts upon objects as they travel through the air. Like all frictional forces, the force of air resistance always Air Resistance opposes the motion of the object. This force will frequently be ignored due to its Force negligible magnitude. It is most noticeable for objects which travel at high Fair speeds (e.g., a skydiver or a downhill skier) or for objects with large surface areas. Tension is the force which is transmitted through a string, rope, or wire when it Tensional is pulled tight by forces acting at each end. The tensional force is directed along Force the wire and pulls equally on the objects on either end of the wire. Ftens The spring force is the force exerted by a compressed or stretched spring upon any object which is attached to it. This force acts to restores the object, which Spring Force compresses or stretches a spring, to its rest or equilibrium position. For most Fspring springs (specifically, for those said to obey "Hooke's Law"), the magnitude of the force is directly proportional to the amount of stretch or compression. Mass vs. Weight The force of gravity is a source of much confusion to many students of physics. The mass of an object refers to the amount of matter that is contained by the object; the weight of an object is the force of gravity acting upon that object. Mass is related to "how much stuff is there" and weight is related to the pull of the Earth (or any other planet) upon that stuff. The mass of an object (measured in kg) will be the same no matter where in the universe that object is located. Mass is never altered by location, the pull of gravity, speed or even the existence of other forces. For example, a 2-kg object will have a mass of 2 kg whether it is located on Earth, on the moon, or on Jupiter; its mass will be 2 kg whether it is moving or not (at least for purposes of this study); and its mass will be 2 kg whether it is being pushed or not. On the other hand, the weight of an object (measured in Newtons) will vary according to where in the universe the object is. Weight depends upon which planet is exerting the force and the distance the object is from the planet. Weight, being equivalent to the force of gravity, is dependent upon the value of g (acceleration of gravity). On Earth's surface, g is 9.8 m/s2 (often approximated to 10 m/s2). On the moon's surface, g is 1.7 m/s2. Go to another planet, and there will be another g value. In addition, the g value is inversely proportional to the distance from the center of the planet. So if g were measured at a distance of 400 km above the earth's surface, you would find the value of g to be less than 9.8 m/s2. Newton's Second Law Newton's second law of motion pertains to the behavior of objects for which all existing forces are not balanced. The second law states that the acceleration of an object is dependent upon two variables – the net force acting upon the object and the mass of the object. The acceleration of an object depends directly upon the net force acting upon the object, and inversely upon the mass of the object. As the net force increases, so will the object's acceleration. However, as the mass of the object increases, its acceleration will decrease. Newton's second law of motion can be formally stated as follows: The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object. In terms of an equation, the net force is equal to the product of the object's mass and its acceleration. Fnet = m * a Throughout this lesson, the emphasis has been on the "net force." The acceleration is directly proportional to the "net force;" the "net force" equals mass times acceleration; the acceleration is in the same direction as the "net force;" an acceleration is produced by a "net force." The NET FORCE. It is important to remember this distinction. Do not use the value of "any 'ole force" in the above equation; it is the net force, not any of the individual forces, which is related to acceleration, the net force is the vector sum of all the forces. If all the individual forces acting upon an object are known, then the net force can be determined. The above equation also indicates that a unit of force is equal to a unit of mass multiplied by a unit of acceleration. By substituting standard metric units for force, mass, and acceleration into the above equation, the following unit equivalency can be written: Thus, the definition of the standard metric unit of force is given by the above equation. One Newton is defined as the amount of force required to give a 1-kg mass an acceleration of 1 m/s2. The net force equation (Fnet = m * a) can also be used as a "recipe" for algebraic problem-solving. The table below can be filled by substituting the known variables into the equation, then solving for the unknown quantity. Try it by yourself and then use the pop-up menus to view the answers. Newton's Second Law of Motion Newton's Second Law of Motion Free Fall and Air Resistance Unit 1 stated that all objects (regardless of their mass) free-fall with the same acceleration – 10 m/s2. This acceleration value is so important in physics that it has its own peculiar name – the acceleration of gravity – and its own peculiar symbol – "g." But why do all objects free-fall at the same rate of acceleration regardless of their mass? Is it because they all weigh the same? ... because they all have the same gravity? ... because the air resistance is the same for each? In addition to an exploration of free-fall, the motion of objects which encounter air resistance will also be analyzed. In particular, two questions will be explored: Why do objects which encounter air resistance ultimately reach a terminal velocity? In situations in which there is air resistance, why do massive objects fall faster than less massive objects? To answer the above questions, Newton's second law of motion (Fnet = m*a) will be applied to analyze the motion of objects which are falling under the influence of gravity only (free-fall) and under the dual influence of gravity and air resistance. Free Fall Motion Free-fall is a special type of motion in which the only force acting upon an object is gravity. Objects, which are said to be undergoing free-fall, do not encounter a significant force of air resistance; they are falling under the sole influence of gravity. Under such conditions, all objects will fall with the same rate of acceleration, regardless of their mass. Why? Consider the free-falling motion of a 10-kg rock and a 1-kg rock. If Newton's second law were applied to their falling motion, and if free-body diagrams were constructed, you would see that the 10-kg rock experiences a greater force of gravity. This greater force of gravity would have a direct effect upon the rock's acceleration; thus, based on force alone, you might think that the 10-kg rock would accelerate faster. But acceleration depends upon two factors: force and mass. The 10-kg rock obviously has more mass (or inertia) than the 1-kg rock. This increased mass has an inverse effect upon the rock's acceleration. Thus, the direct effect of greater force on the 10-kg rock is offset by the inverse effect of its greater mass; and so each rock accelerates at the same rate – 10 m/s2. The ratio of force to mass (Fnet/m) is the same for each rock in situations involving free fall; this ratio (Fnet/m) is equivalent to the acceleration of the object. Falling with Air Resistance As an object falls through air, it usually encounters some degree of air resistance. Air resistance is the result of collisions of the object's leading surface with air molecules. The actual amount of air resistance encountered by an object depends upon a variety of factors. The two most common factors which have a direct effect upon the amount of air resistance present are the speed of the object and the cross-sectional area of the object. Increased speeds result in an increased amount of air resistance. Increased cross- sectional areas result in an increased amount of air resistance. Newton's Third Law A force is a push or a pull upon an object which results from its interaction with another object. Forces result from interactions!, some forces result from contact interactions (normal, frictional, tensional, and applied forces are examples of contact forces) and other forces result from action-at-a-distance interactions (gravitational, electrical, and magnetic forces are examples of action-at-a-distance forces). According to Newton, whenever objects A and B interact with each other, they exert forces upon each other. When you sit in your chair, your body exerts a downward force on the chair and the chair exerts an upward force on your body. There are two forces resulting from this interaction — a force on the chair and a force on your body. These two forces are called action and reaction forces and are the subject of Newton's third law of motion. Formally stated, Newton's third law is:"For every action, there is an equal and opposite reaction." The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the force on the first object equals the size of the force on the second object. The direction of the force on the first object is opposite to the direction of the force on the second object. Forces always come in pairs – equal and opposite action-reaction force pairs. A variety of action-reaction force pairs are evident in nature. Consider the propulsion of a fish through the water. A fish uses its fins to push water backwards. But a push on the water will only serve to accelerate the water. In turn, the water reacts by pushing the fish forwards, propelling the fish through the water. The size of the force on the water equals the size of the force on the fish; the direction of the force on the water (backwards) is opposite to the direction of the force on the fish (forwards). For every action, there is an equal (in size) and opposite (in direction) reaction force. Action-reaction force pairs make it possible for fishes to swim. Consider the flying motion of birds. A bird flies by use of its wings. The wings of a bird push air downwards. In turn, the air reacts by pushing the bird upwards. The size of the force on the air equals the size of the force on the bird; the direction of the force on the air (downwards) is opposite to the direction of the force on the bird (upwards). For every action, there is an equal (in size) and opposite (in direction) reaction. Action-reaction force pairs make it possible for birds to fly. Consider the motion of your automobile on your way to school. An automobile is equipped with wheels that spin backwards. As the wheels spin backwards, they push the road backwards. In turn, the road reacts by pushing the wheels forward. The size of the force on the road equals the size of the force on the wheels (or automobile); the direction of the force on the road (backwards) is opposite to the direction of the force on the wheels (forwards). For every action, there is an equal (in size) and opposite (in direction) reaction. Action-reaction force pairs make it possible for automobiles to move. Identifying Action and Reaction Force Pairs According to Newton's third law, for every action force there is an equal (in size) and opposite (in direction) reaction force. Forces always come in pairs — known as "action-reaction force pairs." Identifying and describing action-reaction force pairs is a simple matter of identifying the two interacting objects and making two statements describing who is pushing on whom and in which direction. For example, consider the interaction between a baseball bat and a baseball. The baseball forces the bat to the right (an action); the bat forces the ball to the left (the reaction). Note that the nouns in the sentence describing the action force switch places when describing the reaction force. Motion Characteristics for Circular Motion Speed and Velocity Suppose that you were driving a car with the steering wheel turned in such a manner that your car followed the path of a perfect circle with a constant radius. And suppose that as you drove, your speedometer maintained a constant reading of 10 mi/hr. In such a situation as this, the motion of your car would be described to be experiencing uniform circular motion. Uniform circular motion is the motion of an object in a circle with a constant or uniform speed. Uniform circular motion - circular motion at a constant speed - is one of many forms of circular motion. An object moving in uniform circular motion would cover the same linear distance in each second of time. When moving in a circle, an object traverses a distance around the perimeter of the circle. So if your car were to move in a circle with a constant speed of 5 m/s, then the car would travel 5 meters along the perimeter of the circle in each second of time. The distance of one complete cycle around the perimeter of a circle is known as the circumference. At a uniform speed of 5 m/s, if the circle had a circumference of 5 meters, then it would take the car 1 second to make a complete cycle around the circle. At this uniform speed of 5 m/s, each cycle around the 5-m circumference circle would require 1 second. At 5 m/s, a circle with a circumference of 20 meters could be made in 4 seconds; and at this uniform speed, every cycle around the 20-m circumference of the circle would take the same time period of 4 seconds. This relationship between the circumference of a circle, the time to complete one cycle around the circle, and the speed of the object is merely an extension of the average speed equation The circumference of any circle can be computed using from the radius according to the equation Circumference = 2*pi*Radius Combining these two equations above will lead to a new equation relating the speed of an object moving in uniform circular motion to the radius of the circle and the time to make one cycle around the circle (period). where R represents the radius of the circle and T represents the period. This equation, like all equations, can be used as a algebraic recipe for problem solving. Yet it also can be used to guide our thinking about the variables in the equation relate to each other. For instance, the equation suggests that for objects moving around circles of different radius in the same period, the object traversing the circle of larger radius must be traveling with the greatest speed. In fact, the average speed and the radius of the circle are directly proportional. A twofold increase in radius corresponds to a twofold increase in speed; a threefold increase in radius corresponds to a three--fold increase in speed; and so on. Objects moving in uniform circular motion will have a constant speed. But does this mean that they will have a constant velocity? Speed is a scalar quantity and velocity is a vector quantity. Velocity, being a vector, has both a magnitude and a direction. The magnitude of the velocity vector is merely the instantaneous speed of the object; the direction of the velocity vector is directed in the same direction which the object moves. Since an object is moving in a circle, its direction is continuously changing. At one moment, the object is moving northward such that the velocity vector is directed northward. One quarter of a cycle later, the object would be moving eastward such that the velocity vector is directed eastward. As the object rounds the circle, the direction of the velocity vector is different than it was the instant before. So while the magnitude of the velocity vector may be constant, the direction of the velocity vector is changing. The best word that can be used to describe the direction of the velocity vector is the word tangential. The direction of the velocity vector at any instant is in the direction of a tangent line drawn to the circle at the object's location. (A tangent line is a line which touches the circle at one point but does not intersect it.) To summarize, an object moving in uniform circular motion is moving around the perimeter of the circle with a constant speed. While the speed of the object is constant, its velocity is changing. Velocity, being a vector, has a constant magnitude but a changing direction. The direction is always directed tangent to the circle and as the object turns the circle, the tangent line is always pointing in a new direction. As we proceed through this unit, we will see that these same principles will have a similar extension to noncircular motion. Applications of Circular Motion Newton's Second Law - Revisited Newton's second law states that the acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to the mass of the object. The law is often expressed in the form of the following two equations. The process of analyzing such physical situations in order to determine unknown information is dependent upon the ability to represent the physical situation by means of a free-body diagram. A free-body diagram is a vector diagram which depicts the relative magnitude and direction of all the individual forces which are acting upon the object. In this Lesson, we will use Unit 2 principles (free-body diagrams, Newton's second law equation, etc.) and circular motion concepts in order to analyze a variety of physical situations involving the motion of objects in circles or along curved paths. The mathematical equations discussed in Lesson 1 and the concept of a centripetal force requirement will be applied in order to analyze roller coasters and other amusement park rides, various athletic movements, and other real-world phenomenon. To illustrate how circular motion principles can be combined with Newton's second law to analyze a physical situation, consider a car moving in a horizontal circle on a level surface. The diagram below depicts the car on the left side of the circle. Applying the concept of a centripetal force requirement, we know that the net force acting upon the object is directed inwards. Since the car is positioned on the left side of the circle, the net force is directed rightward. An analysis of the situation would reveal that there are three forces acting upon the object - the force of gravity (acting downwards), the normal force of the pavement (acting upwards), and the force of friction (acting inwards or rightwards). It is the friction force which supplies the centripetal force requirement for the car to move in a horizontal circle. Without friction, the car would turn its wheels but would not move in a circle (as is the case on an icy surface). This analysis leads to the free-body diagram shown at the right. Observe that each force is represented by a vector arrow which points in the specific direction which the force acts; also notice that each force is labeled according to type (Ffrict, Fnorm, and Fgrav). Such an analysis is the first step of any problem involving Newton's second law and a circular motion. Now consider the following two problems pertaining to this physical scenario of the car making a turn on a horizontal surface. Definition and Mathematics of Work Force and mass information were used to determine the acceleration of an object. Acceleration information was subsequently used to determine information about the velocity or displacement of an object after a given period of time. In this manner, Newton's laws serve as a useful scheme for analyzing motion and making predictions about the final state of an object's motion. In this unit, an entirely different scheme will be utilized to analyze the motion of objects. Motion will be approached from the perspective of work and energy. The effect that work has upon the energy of an object (or system of objects) will be investigated; the resulting velocity and/or height of the object can then be predicted from energy information. In order to understand this work-energy approach to the analysis of motion, it is important to first have a solid understanding of a few basic terms. mechanical energy, potential energy, kinetic energy, and power. In physics, work is defined as a force acting upon an object to cause a displacement. There are three key words in this definition - force, displacement, and cause. In order for a force to qualify as having done work on an object, there must be a displacement and the force must cause the displacement. There are several good examples of work which can be observed in everyday life - a horse pulling a plow through the fields, a father pushing a grocery cart down the aisle of a grocery store, a freshman lifting a backpack full of books upon her shoulder, a weightlifter lifting a barbell above her head, a shot-put launching the shot, etc. In each case described here there is a force exerted upon an object to cause that object to be displaced. Read the following five statements and determine whether or not they represent examples of work. Then depress the mouse upon the pop-up menu to view the answers. Mathematically, work can be expressed by the following equation. where F = force, d = displacement, and the angle (theta) is defined as the angle between the force and the displacement vector. Perhaps the most difficult aspect of the above equation is the angle "theta." The angle is not just any 'ole angle, but rather a very specific angle. The angle measure is defined as the angle between the force and the displacement. To gather an idea of its meaning, consider the following three scenarios. Scenario A: A force acts rightward upon an object as it is displaced rightward. In such an instance, the force vector and the displacement vector are in the same direction. Thus, the angle between F and d is 0 degrees. Scenario B: A force acts leftward upon an object which is displaced rightward. In such an instance, the force vector and the displacement vector are in the opposite direction. Thus, the angle between F and d is 180 degrees. Scenario C: A force acts upward upon an object as it is displaced rightward. In such an instance, the force vector and the displacement vector are at right angles to each other. Thus, the angle between F and d is 90 degrees. Let's consider Scenario C above in more detail. Scenario C involves a situation similar to the waiter who carried a tray full of meals above his head by one arm across the room. It was mentioned earlier that the waiter does not do work upon the tray as he carries it across the room. The force supplied by the waiter on the tray is an upward force and the displacement of the tray is a horizontal displacement. As such, the angle between the force and the displacement is 90 degrees. If the work done by the waiter on the tray were to be calculated, then the results would be 0. Regardless of the magnitude of the force and displacement, F*d*cosine 90 degrees is 0 (since the cosine of 90 degrees is 0). A vertical force can never cause a horizontal displacement; thus, a vertical force does not do work on a horizontally displaced object!! The equation for work lists three variables - each variable is associated with the one of the three key words mentioned in the definition of work (force, displacement, and cause). The angle theta in the equation is associated with the amount of force which causes a displacement. As mentioned in a previous unit, when a force is exerted on an object at an angle to the horizontal, only a part of the force contributes to (or causes) a horizontal displacement. Let's consider the force of a chain pulling upwards and rightwards upon Fido in order to drag Fido to the right. It is only the horizontal component of the tensional force in the chain which causes Fido to be displaced to the right. The horizontal component is found by multiplying the force F by the cosine of the angle between F and d. In this sense, the cosine theta in the work equation relates to the cause factor - it selects the portion of the force which actually causes a displacement. Whenever a new quantity is introduced in physics, the standard metric units associated with that quantity are discussed. In the case of work (and also energy), the standard metric unit is the Joule (abbreviated "J"). One Joule is equivalent to one Newton of force causing a displacement of one meter. In other words, The Joule is the unit of work. 1 Joule = 1 Newton * 1 meter 1J = 1 N * m In fact, any unit of force times any unit of displacement is equivalent to a unit of work. Some nonstandard units for work are shown below. Notice that when analyzed, each set of units is equivalent to a force unit times a displacement unit. In summary, work is a force acting upon an object to cause a displacement. When a force acts to cause an object to be displaced, three quantities must be known in order to calculate the amount of work. Those three quantities are force, displacement and the angle between the force and the displacement. Potential Energy An object can store energy as the result of its position. For example, the heavy ram of a pile driver is storing energy when it is held at an elevated position. This stored energy of position is referred to as potential energy. Similarly, a drawn bow is able to store energy as the result of its position. When assuming its usual position (i.e., when not drawn), there is no energy stored in the bow. Yet when its position is altered from its usual equilibrium position, the bow is able to store energy by virtue of its position. This stored energy of position is referred to as potential energy. Potential energy is the stored energy of position possessed by an object. The two examples above illustrate the two forms of potential energy to be discussed in this course - gravitational potential energy and elastic potential energy. Gravitational potential energy is the energy stored in an object as the result of its vertical position (i.e., height). The energy is stored as the result of the gravitational attraction of the Earth for the object. The gravitational potential energy of the heavy ram of a pile driver is dependent on two variables - the mass of the ram and the height to which it is raised. There is a direct relation between gravitational potential energy and the mass of an object; more massive objects have greater gravitational potential energy. There is also a direct relation between gravitational potential energy and the height of an object; the higher that an object is elevated, the greater the gravitational potential energy. These relationships are expressed by the following equation: PEgrav = mass * g * height PEgrav = m * g * h In the above equation, m represents the mass of the object, h represents the height of the object and g represents the acceleration of gravity (approximately 10 m/s/s on Earth). To determine the gravitational potential energy of an object, a zero height position must first be arbitrarily assigned. Typically, the ground is considered to be a position of zero height. But this is merely an arbitrarily assigned position which most people agree upon. Since many of our labs are done on tabletops, it is often customary to assign the tabletop to be the zero height position; again this is merely arbitrary. If the tabletop is the zero position, then the potential energy of an object is based upon its height relative to the tabletop. For example, a pendulum bob swinging to and from above the table top has a potential energy which can be measured based on its height above the tabletop. By measuring the mass of the bob and the height of the bob above the tabletop, the potential energy of the bob can be determined. Since the gravitational potential energy of an object is directly proportional to its height above the zero position, a doubling of the height will result in a doubling of the gravitational potential energy. A tripling of the height will result in a tripling of the gravitational potential energy. Use this principle to determine the blanks in the following diagram. The second form of potential energy which we will discuss in this course is elastic potential energy. Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing. Elastic potential energy can be stored in rubber bands, bungee chords, trampolines, springs, an arrow drawn into a bow, etc. The amount of elastic potential energy stored in such a device is related to the amount of stretch of the device - the more stretch, the more stored energy. Springs are a special instance of a device which can store elastic potential energy due to either compression or stretching. A force is required to compress a spring; the more compression there is, the more force which is required to compress it further. For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x); the constant of proportionality is known as the spring constant (k). Such springs are said to follow Hooke's Law. If a spring is not stretched or compressed, then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally assumes when there is no force applied to it. In terms of potential energy, the equilibrium position could be called the zero-potential energy position. There is a special equation for springs which relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant. The equation is To summarize, potential energy is the energy which an object has stored due to its position relative to some zero position. An object possesses gravitational potential energy if it is positioned at a height above (or below) the zero height position. An object possesses elastic potential energy if it is at a position on an elastic medium other than the equilibrium position. Kinetic Energy Kinetic energy is the energy of motion. An object which has motion - whether it be vertical or horizontal motion - has kinetic energy. There are many forms of kinetic energy - vibrational (the energy due to vibrational motion), rotational (the energy due to rotational motion), and translational (the energy due to motion from one location to another). To keep matters simple, we will focus upon translational kinetic energy. The amount of translational kinetic energy (from here on, the phrase kinetic energy will refer to translational kinetic energy) which an object has depends upon two variables: the mass (m) of the object and the speed (v) of the object. The following equation is used to represent the kinetic energy (KE) of an object. where m = mass of object v = speed of object This equation reveals that the kinetic energy of an object is directly proportional to the square of its speed. That means that for a twofold increase in speed, the kinetic energy will increase by a factor of four; for a threefold increase in speed, the kinetic energy will increase by a factor of nine; and for a fourfold increase in speed, the kinetic energy will increase by a factor of sixteen. The kinetic energy is dependent upon the square of the speed. As it is often said, an equation is not merely a recipe for algebraic problem-solving, but also a guide to thinking about the relationship between quantities. Kinetic energy is a scalar quantity; it does not have a direction. Unlike velocity, acceleration, force, and momentum, the kinetic energy of an object is completely described by magnitude alone. Like work and potential energy, the standard metric units of measurement for kinetic energy is the Joule. As might be implied by the above equation, 1 Joule is equivalent to 1 kg*(m/s)^2. Mechanical Energy It was said that work is done upon an object whenever a force acts upon it to cause it to be displaced. Work is a force acting upon an object to cause a displacement. In all instances in which work is done, there is an object which supplies the force in order to do the work. If a World Civilization book is lifted to the top shelf of a student locker, then the student supplies the force to do the work on the book. If a plow is displaced across a field, then some form of farm equipment (usually a tractor or a horse) supplies the force to do the work on the plow. If a pitcher winds up and accelerates a baseball towards home plate, then the pitcher supplies the force to do the work on the baseball. If a roller coaster car is displaced from ground level to the top of the first drop of the Shock Wave, then a chain (driven by a motor) supplies the force to do the work on the car. If a barbell is displaced from ground level to a height above a weightlifter's head, then the weightlifter is supplying a force to do work on the barbell. In all instances, an object which possesses some form of energy supplies the force to do the work. In the instances described here, the objects doing the work (a student, a tractor, a pitcher, a motor/chain) possess chemical potential energy stored in food or fuel which is transformed into work. In the process of doing work, the objects doing the work exchange energy in one form to do work on another object to give it energy. The energy acquired by the objects upon which work is done is known as mechanical energy. Mechanical energy is the energy which is possessed by an object due to its motion or its stored energy of position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position). Objects have mechanical energy if they are in motion and/or if they are at some position relative to a zero potential energy position (for example, a brick held at a vertical position above the ground or zero height position). A moving car possesses mechanical energy due to its motion (kinetic energy). A moving baseball possesses mechanical energy due to both its high speed (kinetic energy) and its vertical position above the ground (gravitational potential energy). A World Civilization book at rest on the top shelf of a locker possesses mechanical energy due to its vertical position above the ground (gravitational potential energy). A barbell lifted high above a weightlifter's head possesses mechanical energy due to its vertical position above the ground (gravitational potential energy). A drawn bow possesses mechanical energy due to its stretched position (elastic potential energy). An object which possesses mechanical energy is able to do work. In fact, mechanical energy is often defined as the ability to do work. Any object which possesses mechanical energy - whether it be in the form of potential energy or kinetic energy - is able to do work. That is, its mechanical energy enables that object to apply a force to another object in order to cause it to be displaced. Numerous examples can be given of how an object with mechanical energy can harness that energy in order to apply a force to cause another object to be displaced. A classic example involves the heavy ram of a pile driver. A pile driver consists of a massive object which is elevated to a high position and allowed to fall upon another object (called the pile) in order to drive it downwards. Upon hitting the pile, the ram applies a force to it in order to cause it to be displaced. The diagram below depicts the process by which the mechanical energy of a pile driver can be used to do work. A hammer is a miniature version of a pile driver. The mechanical energy of a hammer gives the hammer its ability to apply a force to a nail in order to cause it to be displaced. Because the hammer has mechanical energy (in the form of kinetic energy), it is able to do work on the nail. Mechanical energy is the ability to do work. Another example which illustrates how mechanical energy is the ability of an object to do work can be seen any evening at your local bowling alley. The mechanical energy of a bowling ball gives the ball the ability to apply a force to a bowling pin in order to cause it to be displaced. Because the massive ball has mechanical energy (in the form of kinetic energy), it is able to do work on the pin. Mechanical energy is the ability to do work. A dart gun is still another example of how mechanical energy of an object can do work on another object. When a dart gun is loaded and the springs are compressed, it possesses mechanical energy. The mechanical energy of the compressed springs give the springs the ability to apply a force to the dart in order to cause it to be displaced. Because of the springs have mechanical energy (in the form of elastic potential energy), it is able to do work on the dart. Mechanical energy is the ability to do work. A common scene in the western United States is a "wind farm." High speed winds are used to do work on the blades of a turbine at the so-called wind farm. The mechanical energy of the moving air give the air the ability to apply a force and cause a displacement of the blades. As the blades spin, their energy is subsequently converted into electrical energy (a non-mechanical form of energy) and supplied to homes and industries in order to run electrical appliances. Because the moving wind has mechanical energy (in the form of kinetic energy), it is able to do work on the blades. Once more, mechanical energy is the ability to do work. As already mentioned, the mechanical energy of an object can be the result of its motion (i.e., kinetic energy) and/or the result of its stored energy of position (i.e., potential energy). The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy. This sum is simply referred to as the total mechanical energy (abbreviated TME). TME = PE + KE As discussed earlier, there are two forms of potential energy discussed in our course - gravitational potential energy and elastic potential energy. Given this fact, the above equation can be rewritten: TME = PEgrav + PEspring + KE The diagram below depicts the motion of Li Ping Phar (esteemed Chinese ski jumper) as she glides down the hill and makes one of her record-setting jumps. The total mechanical energy of Li Ping Phar is the sum of the potential and kinetic energies. The two forms of energy sum up to 50 000 Joules. Notice also that the total mechanical energy of Li Ping Phar is a constant value throughout her motion. There are conditions under which the total mechanical energy will be a constant value and conditions under which it is a changing value. This is the subject of Lesson 2 - the work-energy theorem. For now, merely remember that total mechanical energy is the energy possessed by an object due to either its motion or its stored energy of position. The total amount of mechanical energy is merely the sum of these two forms of energy. And finally, an object with mechanical energy is able to do work on another object. Power The quantity work has to do with a force causing a displacement. Work has nothing to do with the amount of time that this force acts to cause the displacement. Sometimes, the work is done very quickly and other times the work is done rather slowly. For example, a rock climber takes an abnormally long time to elevate her body up a few meters along the side of a cliff. On the other hand, a trail hiker (who selects the easier path up the mountain) might elevate her body a few meters in a short amount of time. The two people might do the same amount of work, yet the hiker does the work in considerably less time than the rock climber. The quantity which has to do with the rate at which a certain amount of work is done is known as the power. The hiker has a greater power rating than the rock climber. Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation. The standard metric unit of power is the Watt. As is implied by the equation for power, a unit of power is equivalent to a unit of work divided by a unit of time. Thus, a Watt is equivalent to a Joule/second. For historical reasons, the horsepower is occasionally used to describe the power delivered by a machine. One horsepower is equivalent to approximately 750 Watts. Most machines are designed and built to do work on objects. All machines are typically described by a power rating. The power rating indicates the rate at which that machine can do work upon other objects. Thus, the power of a machine is the work/time ratio for that particular machine. A car engine is an example of a machine which is given a power rating. The power rating relates to how rapidly the car can accelerate the car. Suppose that a 40-horsepower engine could accelerate the car from 0 mi/hr to 60 mi/hr in 16 seconds. If this were the case, then a car with four times the horsepower could do the same amount of work in one-fourth the time. That is, a 160- horsepower engine could accelerate the same car from 0 mi/hr to 60 mi/hr in 4 seconds. The point is that for the same amount of work, power and time are inversely proportional. Equations can be "guides to thinking" about the relationships between quantities. The power equation suggests that a more powerful engine can do the same amount of work in less time. A person is also a machine which has a power rating. Some people are more power-full than others; that is, they are capable of doing the same amount of work in less time or more work in the same amount of time. In the Personal Power lab, students determined their own personal power by doing work on their bodies to elevate it up a flight of stairs. By measuring the force, displacement and time, we were able to measure our personal power rating. Suppose that Ben Pumpiniron elevates his 80-kg body up the 2.0 meter stairwell in 1.8 seconds. If this were the case, then we could calculate Ben's power rating. It can be assumed that Ben must apply a 800-Newton downward force upon the stairs to elevate his body. By so doing, the stairs would push upward on Ben's body with just enough force to lift his body up the stairs. It can also be assumed that the angle between the force of the stairs on Ben and Ben's displacement is 0 degrees. With these two approximations, Ben's power rating could be determined as shown below. Ben's power rating is 889 Watts; what a "horse." The expression for power is work/time. Now since the expression for work is force*displacement, the expression for power can be rewritten as (force*displacement)/time. Yet since the expression for velocity is displacement/time, the expression for power can be rewritten once more as force*velocity. This is shown below. This new expression for power reveals that a powerful machine is both strong (big force) and fast (big velocity). The powerful car engine is strong and fast. The powerful farm equipment is strong and fast. The powerful weightlifters are strong and fast. The powerful linemen on a football team are strong and fast. A machine which is strong enough to apply a big force to cause a displacement in a small mount of time (i.e., a big velocity) is a powerful machine. The Impulse-Momentum Change Theorem Momentum The sports announcer says "Going into the all-star break, the Chicago White Sox have the momentum." The headlines declare "Chicago Bulls Gaining Momentum." The coach pumps up his team at half-time, saying "You have the momentum; the critical need is that you use that momentum and bury them in this third quarter." Momentum is a commonly used term in sports. A team that has the momentum is on the move and is going to take some effort to stop. A team that has a lot of momentum is really on the move and is going to be hard to stop. Momentum is a physics term; it refers to the quantity of motion that an object has. A sports team which is "on the move" has the momentum. If an object is in motion ("on the move") then it has momentum. Momentum can be defined as "mass in motion." All objects have mass; so if an object is moving, then it has momentum - it has its mass in motion. The amount of momentum which an object has is dependent upon two variables: how much stuff is moving and how fast the stuff is moving. Momentum depends upon the variables mass and velocity. In terms of an equation, the momentum of an object is equal to the mass of the object times the velocity of the object. Momentum = mass * velocity In physics, the symbol for the quantity momentum is the small case "p"; thus, the above equation can be rewritten as p=m*v where m = mass and v=velocity. The equation illustrates that momentum is directly proportional to an object's mass and directly proportional to the object's velocity. The units for momentum would be mass units times velocity units. The standard metric unit of momentum is the kg*m/s. While the kg*m/s is the standard metric unit of momentum, there are a variety of other units which are acceptable (though not conventional) units of momentum; examples include kg*mi/hr, kg*km/hr, and g*cm/s. In each of these examples, a mass unit is multiplied by a velocity unit to provide a momentum unit. This is consistent with the equation for momentum. Momentum is a vector quantity. As discussed in an earlier unit, a vector quantity is a quantity which is fully described by both magnitude and direction. To fully describe the momentum of a 5- kg bowling ball moving westward at 2 m/s, you must include information about both the magnitude and the direction of the bowling ball. It is not enough to say that the ball has 10 kg*m/s of momentum; the momentum of the ball is not fully described until information about its direction is given. The direction of the momentum vector is the same as the direction of the velocity of the ball. In a previous unit, it was said that the direction of the velocity vector is the same as the direction which an object is moving. If the bowling ball is moving westward, then its momentum can be fully described by saying that it is 10 kg*m/s, westward. As a vector quantity, the momentum of an object is fully described by both magnitude and direction. From the definition of momentum, it becomes obvious that an object has a large momentum if either its mass or its velocity is large. Both variables are of equal importance in determining the momentum of an object. Consider a Mack truck and a roller skate moving down the street at the same speed. The considerably greater mass of the Mack truck gives it a considerably greater momentum. Yet if the Mack truck were at rest, then the momentum of the least massive roller skate would be the greatest; for the momentum of any object which is at rest is 0. Objects at rest do not have momentum - they do not have any "mass in motion." Both variables - mass and velocity - are important in comparing the momentum of two objects. The Law of Momentum Conservation The Law of Action-Reaction (Revisited) A collision is an interaction between two objects which have made contact (usually) with each other. As in any interaction, a collision results in a force being applied to the two colliding objects. Such collisions are governed by Newton's laws of motion. InNewton's third law of motion was introduced and discussed. It was said that... ... in every interaction, there is a pair of forces acting on the two interacting objects. The size of the force on the first object equals the size of the force on the second object. The direction of the force on the first object is opposite to the direction of the force on the second object. Forces always come in pairs - equal and opposite action-reaction force pairs. Newton's third law of motion is naturally applied to collisions between two objects. In a collision between two objects, both objects experience forces which are equal in magnitude and opposite in direction. Such forces cause one object to speed up (gain momentum) and the other object to slow down (lose momentum). According to Newton's third law, the forces on the two objects are equal in magnitude. While the forces are equal in magnitude and opposite in direction, the acceleration of the objects are not necessarily equal in magnitude. In accord with Newton's second law of motion, the acceleration of an object is dependent upon both force and mass. Thus, if the colliding objects have unequal mass, they will have unequal accelerations as a result of the contact force which results during the collision. Consider the collision between the club head and the golf ball in the sport of golf. When the club head of a moving golf club collides with a golf ball at rest upon a tee, the force experienced by the club head is equal to the force experienced by the golf ball. Most observers of this collision have difficulty with this concept because they perceive the high speed given to the ball as the result of the collision. They are not observing unequal forces upon the ball and club head, but rather unequal accelerations. Both club head and ball experience equal forces, yet the ball experiences a greater acceleration due to its smaller mass. In a collision, there is a force on both objects which causes an acceleration of both objects; the forces are equal in magnitude and opposite in direction, yet the least massive object receives the greatest acceleration. Consider the collision between a moving seven-ball and an eight-ball that is at rest in the sport of billiards. When the seven-ball collides with the eight-ball, each ball experiences an equal force directed in opposite directions. The rightward moving seven-ball experiences a leftward force which causes it to slow down; the eight-ball experiences a rightward force which causes it to speed up. Since the two balls have equal masses, they will also experience equal accelerations. In a collision, there is a force on both objects which causes an acceleration of both objects; the forces are equal in magnitude and opposite in direction. For collisions between equal-mass objects, each object experiences the same acceleration. Consider the interaction between a male and female figure skater in pair figure skating. A woman (m = 45 kg) is kneeling on the shoulders of a man (m = 70 kg); the pair is moving along the ice at 1.5 m/s. The man gracefully tosses the woman forward through the air and onto the ice. The woman receives the forward force and the man receives a backward force. The force on the man is equal in magnitude and opposite in direction to the force on the woman. Yet the acceleration of the woman is greater than the acceleration of the man due to the smaller mass of the woman. Many observers of this interaction have difficulty believing that the man experienced a backward force. "After all," they might argue, "the man did not move backward." Such observers are presuming that forces cause motion; that is a backward force would cause a backward motion. This is a common misconception that has been addressed elsewhe Forces cause acceleration, not motion. The male figure skater experiences a backwards (you might say "negative") force which causes his backwards (or "negative") acceleration; that is, the man slowed down while the woman sped up. In every interaction (with no exception), there are forces acting upon the two interacting objects which are equal in magnitude and opposite in direction. Collisions are governed by Newton's laws. The law of action-reaction (Newton's third law) explains the nature of the forces between the two interacting objects. According to the law, the force exerted by object 1 upon object 2 is equal in magnitude and opposite in direction to the force exerted by object 2 upon object 1. Waves and wavelike Motion Waves are everywhere. Whether we recognize or not, we encounter waves on a daily basis. Sound waves, visible light waves, radio waves, microwaves, water waves, sine waves, cosine waves, telephone chord waves, stadium waves, earthquake waves, waves on a string, and slinky waves and are just a few of the examples of our daily encounters with waves. In addition to waves, there are a variety of phenomenon in our physical world which resemble waves so closely that we can describe such phenomenon as being wavelike. The motion of a pendulum, the motion of a mass suspended by a spring, the motion of a child on a swing, and the "Hello, Good Morning!" wave of the hand can be thought of as wavelike phenomena. Waves (and wavelike phenomena) are everywhere! We study the physics of waves because it provides a rich glimpse into the physical world which we seek to understand and describe as physicists. Before beginning a formal discussion of the nature of waves, it is often useful to ponder the various encounters and exposures which we have of waves. Where do we see waves or examples of wavelike motion? What experiences do we already have which will help us in understanding the physics of waves? For many people, the first thought concerning waves conjures up a picture of a wave moving across the surface of an ocean, lake, pond or other body of water. The waves are created by some form of a disturbance, such as a rock thrown into the water, a duck shaking its tail in the water or a boat moving through the water. The water wave has a crest and a trough and travels from one location to another. One crest is often followed by a second crest which is often followed by a third crest. Every crest is separated by a trough to create an alternating pattern of crests and troughs. A duck or gull at rest on the surface of the water is observed to bob up-and-down at rather regular time intervals as the wave passes by. The waves may appear to be plane waves which travel together as a front in a straight-line direction, perhaps towards a sandy shore. Or the waves may be circular waves which originate from the point where the disturbances occur; such circular waves travel across the surface of the water in all directions. These mental pictures of water waves are useful for understanding the nature of a wave and will be revisited later when we begin our formal discussion of the topic. Finally, we are familiar with microwaves and visible light waves. While we have never seen them, we believe that they exist because we have witnessed how they carry energy from one location to another. And similarly, we are familiar with radio waves and sound waves. Like microwaves, we have never seen them. Yet we believe they exist because we have witnessed the signals which they carry from one location to another and we have even learned how to tune into those signals through use of our ears or a tuner on a television or radio. Waves, as we will learn, carry energy from one location to another. And if the frequency of those waves can be changed, then we can also carry a complex signal which is capable of transmitting an idea or thought from one location to another. Waves are everywhere in nature. Our understanding of the physical world is not complete until we understand the nature, properties and behaviors of waves. The goal of this unit is to develop mental models of waves and ultimately apply those models to an understanding of the two most common types of waves - sound waves and light waves. Properties of Waves The Anatomy of a Wave A transverse wave is a wave in which the particles of the medium are displaced in a direction perpendicular to the direction of energy transport. A transverse wave can be created in a rope if the rope is stretched out horizontally and the end is vibrated back-and-forth in a vertical direction. If a snap-shot of such a transverse wave could be taken so as to freeze the shape of the rope in time, then it would look like the following diagram. The dashed line drawn through the center of the diagram represents the equilibrium or rest position of the string. This is the position that the string would assume if there were no disturbance moving through it. Once a disturbance is introduced into the string, the particles of the string begin to vibrate upwards and downwards. At any given moment in time, a particle on the medium could be above or below the rest position. Points A and F on the diagram represent the crests of this wave. The crest of a wave is the point on the medium which exhibits the maximum amount of positive or upwards displacement from the rest positon. Points D and I on the diagram represent the troughs of this wave. The trough of a wave is the point on the medium which exhibits the maximum amount of negative or downwards displacement from the rest positon. The wave shown above can be described by a variety of properties. One such property is amplitude. The amplitude of a wave refers to the maximum amount of displacement of a a particle on the medium from its rest position. In a sense, the amplitude is the distance from rest to crest. Similarly, the amplitude can be measured from the rest position to the trough position. In the diagram above, the amplitude could be measured as the distance of a line segment which is perpendicular to the rest position and extends vertically upward from the rest position to point A. The wavelength is another property of a wave which is portrayed in the diagram above. The wavelength of a wave is simply the length of one complete wave cycle. If you were to trace your finger across the wave in the diagram above, you would notice that your finger repeats its path. A wave has a repeating pattern. And the length of one such repetition (known as a wave cylce) is the wavelength. The wavelength can be measured as the distance from crest to crest or from trough to trough. In fact, the wavelength of a wave can be measured as the distance from a point on a wave to the corresponding point on the next cycle of the wave. In the diagram above, the wavelength is the distance from A to E, or the distance from B to G, or the distance from E to J, or the distance from D to I, or the distance from C to H. Any one of these distance measurements would suffice in determining the wavelength of this wave. A longitudinal wave is a wave in which the particles of the medium are displaced in a direction parallel to the direction of energy transport. A longitudinal wave can be created in a slinky if the slinky is stretched out horizontally and the end coil is vibrated back-and-forth in a horizontal direction. If a snap-shot of such a longitudinal wave could be taken so as to freeze the shape of the slinky in time, then it would look like the following diagram. \ Because the coils of the slinky are vibrating longitudinally, there are regions where they become pressed together and other regions where they are spread apart. A region where the coils are pressed together in a small amount of space is known as a compression. A compression is a point on a medium through which a longitudinal wave is traveling which has the maximum density. A region where the coils are spread apart, thus maximizing the distance between coils, is known as a rarefaction. A rarefaction is a point on a medium through which a longitudinal wave is traveling which has the minimum density. Points A, C and E on the diagram above represent compressions and points B, D, and F represent rarefactions. While a transverse wave has an alternating pattern of crests and troughs, a longitudinal wave has an alternating pattern of compressions and rarefactions. As discussed above, the wavelength of a wave is the length of one complete cycle of a wave. For a transverse wave, the wavelength is determined by measuring from crest to crest. A longitudinal wave does not have crest; so how can its wavelength be determined? The wavelength can always be determined by measuring the distance between any two corresponding points on adjacent waves. In the case of a longitudinal wave, a wavelength measurement is made by measuring the distance from a compression to the next compression or from a rarefaction to the next rarefaction. On the diagram above, the distance from point A to point C or from point B to point D would be representative of the wavelength. Behavior of Waves Wave-like nature of light. A wave doesn't just stop when it reaches the end of the medium. Rather, a wave will undergo certain behaviors when it encounters the end of the medium. Specifically, there will be some reflection off the boundary and some transmission into the new medium. The transmitted wave undergoes refraction (or bending) if it approaches the boundary at an angle. If the boundary is merely an obstacle implanted within the medium, and if the dimensions of the obstacle are smaller than the wavelength of the wave, then there will be very noticeable diffraction of the wave around the object. Each one of these behaviors - reflection, refraction and diffraction - is characterized by specific conceptual principles and mathematical equations. The reflection, refraction, and diffraction of waves was first introduced in Now we will see how light waves demonstrate their wave nature by reflection, refraction and diffraction. All waves are known to undergo reflection or the bouncing off of an obstacle. Most people are very accustomed to the fact that light waves also undergo reflection. The reflection of light waves off of a mirrored surface results in the formation of an image. One characteristic of wave reflection is that the angle at which the wave approaches a flat reflecting surface is equal to the angle at which the wave leaves the surface. This characteristic is observed for water waves and sound waves. It is also observed for light waves. Light, like any wave, follows the law of reflection when bouncing off flat surfaces. For now, it is enough to say that the reflective behavior of light provides evidence for the wave-like nature of light. All waves are known to undergo refraction when they pass from one medium to another medium. That is, when a wavefront crosses the boundary between two media, the direction that the wavefront is moving undergoes a sudden change; the path is "bent." This behavior of wave refraction can be described by both conceptual and mathematical principles. First, the direction of "bending" is dependent upon the relative speed of the two media. A wave will bend one way when it passes from a medium in which it travels slow into a medium in which it travels fast; and if moving from a fast medium to a slow medium, the wavefront will bend in the opposite direction. Second, the amount of bending is dependent upon the actual speeds of the two media on each side of the boundary. The amount of bending is a measurable behavior which follows distinct mathematical equations. These equations are based upon the speeds of the wave in the two media and the angles at which the wave approaches and departs from the boundary. Light, like any wave, is known to refract as it passes from one medium into another medium. In fact, a study of the refraction of light reveals that its refractive behavior follows the same conceptual and mathematical rules which govern the refractive behavior of other waves such as water waves and sound waves. For now, it is enough to say that the refractive behavior of light provides evidence for the wave-like nature of light. Reflection involves a change in direction of waves when they bounce off a barrier; refraction of waves involves a change in the direction of waves as they pass from one medium to another; and diffraction involves a change in direction of waves as they pass through an opening or around an obstacle in their path. Water waves have the ability to travel around corners, around obstacles and through openings. Sound waves do the same. But what about light? Do light waves bend around obstacles and through openings? If they do, then it would provide still more evidence to support the belief that light is a wave. When light encounters an obstacle in its path, the obstacle blocks the light and tends to cause the formation of a shadow in the region behind the obstacle. Light does not exhibit a very noticeable ability to bend around the obstacle and fill in the region behind it with light. Nonetheless, light does diffract around obstacles. In fact, if you observe a shadow carefully, you will notice that its edges are extremely fuzzy. Interference effects occur due to the diffraction of light around different sides of the object, causing the shadow of the object to be fuzzy. This was demonstrated in class with a laser light and penny demonstration. Light diffracting around the right edge of a penny can constructively and destructively interfere with light diffracting around the left edge of the penny. The result is that an interference pattern is created; the pattern consists of alternating rings of light and darkness. Such a pattern is only noticeable if a narrow beam of monochromatic light (i.e., single wavelength light) is passed directed at the penny. The photograph at the right shows an interference pattern created in this manner. Since, light waves are diffracting around the edges of the penny, the waves are broken up into different wavefronts which converge at a point on a screen to produce the interference pattern shown in the photograph. This amazing penny diffraction demonstration provides another reason why believing that light has a wave-like nature Light behaves as a wave - it undergoes reflection, refraction, and diffraction just like any wave would. How Do We Know Light Behaves as a Wave? Coulomb's Law Like charges repel, unlike charges attract. Coulomb's law states that the electrical force between two charged objects is directly proportional to the product of the quantity of charge on the objects and inversely proportional to the square of the separation distance between the two objects. In equation form, Coulomb's law can be stated as where Q1 represents the quantity of charge on object 1 (in Coulombs), Q2 represents the quantity of charge on object 2 (in Coulombs), and d represents the distance of separation between the two objects (in meters). The symbol k is a proportionality constant known as the Coulomb's law constant Electric Charge The unit of electric charge is the coulomb. Ordinary matter is made up of atoms which have positively charged nuclei and negatively charged electrons surrounding them. Charge is quantized as a multiple of the electron or proton charge: The influence of charges is characterized in terms of the forces between them (Coulomb's law) and the electric field and voltage produced by them. One coulomb of charge is the charge which would flow through a 120 watt lightbulb (120 volts AC) in one second. Two charges of one coulomb each separated by a meter would repel each other with a force of about a million tons! The rate of flow of electric charge is called electric current and is measured in amperes. Electric Current Electric current is the rate of charge flow past a given point in an electric circuit, measured in coulombs/second which is named amperes. In most DC electric circuits, it can be assumed that the resistance to current flow is a constant so that the current in the circuit is related to voltage and resistance by Ohm's law. Voltage Voltage is electric potential energy per unit charge, measured in joules per coulomb ( = volts). It is often referred to as "electric potential", which then must be distinguished from electric potential energy by noting that the "potential" is a "per-unit-charge" quantity. Like mechanical potential energy, the zero of potential can be chosen at any point, so the difference in voltage is the quantity which is physically meaningful. The difference in voltage measured when moving from point A to point B is equal to the work which would have to be done, per unit charge, against the electric field to move the charge from A to B. Used to express Used to calculate the Used to calculate current conservation of energy Is generated by moving a potential from a in Ohm's law. around a circuit in the wire in a magnetic fi distribution of charges. voltage law. Ohm's Law For many conductors of electricity, the electric current which will flow through them is directly proportional to the voltage applied to them. When a microscopic view of Ohm's law is taken, it is found to depend upon the fact that the drift velocity of charges through the material is proportional to the electric field in the conductor. The ratio of voltage to current is called the resistance, and if the ratio is constant over a wide range of voltages, the material is said to be an "ohmic" material. If the material can be characterized by such a resistance, then the current can be predicted from the relationship: Current Law The electric current in amperes which flows into any junction in an electric circuit is equal to the current which flows out. This can be seen to be just a statement of conservation of charge. Since you do not lose any charge during the flow process around the circuit, the total current in any cross-section of the circuit is the same. Along with the voltage law, this law is a powerful tool for the analysis of electric circuits. Voltage Law The voltage changes around any closed loop must sum to zero. No matter what path you take through an electric circuit, if you return to your starting point you must measure the same voltage, constraining the net change around the loop to be zero. Since voltage is electric potential energy per unit charge, the voltage law can be seen to be a consequence of conservation of energy. The voltage law has great practical utility in the analysis of electric circuits. It is used in conjunction with the current law in many circuit analysis tasks. Resistor Combinations The combination rules for any number of resistors in series or parallel can be derived with the use of Ohm's Law, the voltage law, and the current law. Kirchhoff's Current Law This fundamental law results from the conservation of charge. It applies to a junction or node in a circuit -- a point in the circuit where charge has several possible paths to travel. In Figure 1, we see that IA is the only current flowing into the node. However, there are three paths for current to leave the node, and these current are represented by IB, IC, and ID. Once charge has entered into the node, it has no place to go except to leave (this is known as conservation of charge). The total charge flowing into a node must be the same as the the total charge flowing out of the node. So, IB + IC + ID = IA Bringing everything to the left side of the above equation, we get (IB + IC + ID) - IA = 0 Figure 1 Possible node (or junction) in a circuit Then, the sum of all the currents is zero. This can be generalized as follows

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