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Chapter 4- Newton’s laws of Motion • Kinematics – the study of how objects move • Dynamics – the study of why objects move • Force – a push or a pull; symbol is F; SI unit is the Newton, or N – One Newton is the force necessary to cause a one kilogram mass to accelerate at the rate of 1 m/s2 – 1 N = 1 kg m/s2 • Statics – Statics is the study of the forces acting on and within structures that are in equilibrium. Statics means that the body is at rest. • Conditions for static equilibrium – The sum of the forces acting on the body must add up to zero. (Remember, if a force component points in the negative x, y, or z direction, it is assigned a negative value. SFx = 0 SFy = 0 SFz = 0 – The sum of all the torques acting on the object (calculated along any axis) must be zero. St = 0 • Four basic forces: – Gravitational force -an attractive force that exists between all objects with mass; an object with mass attracts another object with mass; the magnitude of the force is directly proportional to the masses of the two objects and inversely proportional to the square of the distance between the two objects. Stated mathematically: – Where G is the universal gravitational constant (meaning it has the same value throughout the universe), m1 and m2 are the masses of the objects in kilograms, and d is the distance between them in meters – G = 6.67 x 10-11 N m2 / kg2 – Cavendish found the universal gravitation constant, allowing the earth to be "weighed." – Check out Black Holes, Gravitational Waves, and Escape Velocity under Advanced Newton’s Laws of Motion: • Newton’s First Law – an object in motion remains in motion at constant speed and moving in a straight line and an object at rest remains at rest unless acted upon by an outside force; called the law of inertia. • You do not need a force to keep a body moving at a constant velocity. Also, a body at rest in one frame of reference could be moving at constant speed relative to another frame of reference. – Equilibrium an object is in equilibrium when its velocity is zero or its velocity is constant – Inertia a measure of how an object resists changes in motion; it is a measure of an object’s mass Electromagnetic force an attractive or a repulsive force between charged particles; when charged particles are in motion, they produce magnetic forces on each other; the magnitude of the force is directly proportional to the magnitudes of the charges and inversely proportional to the square of the distance between the two charges Strong nuclear force an attractive force between the particles in the nucleus; it is the strongest of the four forces, but only acts over very small distances (does not obey inverse square law for distances) Weak nuclear force a force involved in the radioactive decay of some nuclei (in the 1960’s, Weinberg theorized the existence of the electroweak force, combing the electromagnetic and the weak nuclear force) • Newton’s Second Law – An unbalanced force (or net force, S) causes an object to accelerate; this acceleration is directly proportional to the unbalanced force and inversely proportional to the object’s mass; called the law of acceleration – a = SF / m or S F = m a – Equilibrium • an object is in equilibrium when no unbalanced force (or net force) acts on it – Mass • A measure of the inertia of a body. (SI unit is the kilogram); Mass is a property of a body itself. It should not be confused with weight, which is a force (the force of gravity acting on an object). – An excellent Newton's Second Law Applet that allows you to conduct a simulated Second Law experiment. • Newton’s Third Law – When one object exerts a force on another object, the second object exerts a force on the first object that is equal in magnitude, but opposite in direction; for every action, there is an equal, but opposite reaction; called the law of action-reaction • A good site that tests your understanding of action/reaction pairs of forces Advanced Look at the 3rd Law • Where do forces come from? A force is exerted on an object and is exerted by another object. The "action" force acts on a different body than a "reaction" force. To experience the 3rd law, push on a desk with your hand. Your hand's shape is distorted, showing that a force is acting on it. You can feel the force that the desk exerts on your hand. The harder that you push on the desk, the harder that the desk pushes back on your hand. You only feel the forces exerted on you; you do not feel the forces that you exert on something else. Applications of Newton's Laws • Free-body diagrams represent forces (their magnitudes and their directions). In a free-body diagram, all forces are represented using arrows. The direction of motion is considered positive (usually assigned the right direction); the direction opposite the motion is negative (typically assigned the left direction); up is positive and down is negative • If the sum of all the horizontal forces acting on an object is zero, then the object is in equilibrium horizontally. If the sum of all the vertical forces acting on an object is zero, then the object is in equilibrium vertically. If the sum of the forces in a direction is not zero, then the forces in that direction are not balanced. We say that an unbalanced force acts in that direction. An object can be in equilibrium in one direction and not in equilibrium in another direction. Forces that act horizontally are independent of forces that act vertically. To work free-body diagram problems: • Draw the free-body diagram labeling all forces (their magnitudes and directions). Remember to use the appropriate positive and negative signs. • Add all the forces in the vertical direction. If the sum is zero, the forces are balanced, and there is no acceleration. If the sum is not zero, the forces are unbalanced, and there is acceleration in the vertical direction. This sum equals the product of mass times acceleration. • Add all the forces in the horizontal direction. If the sum is zero, the forces are balanced, and there is no acceleration. If the sum is not zero, the forces are unbalanced, and there is acceleration in the horizontal direction. This sum equals the product of mass times acceleration. Common types of forces involved in motion: 1. Weight – a measure of the gravitational force acting on an object; direction is down (toward the earth’s center); symbol is Fg – Fg = m g – Where Fg is the weight of the object in Newtons, m is the mass of the object in kilograms, and g is the acceleration due to gravity. – Letting the gravitational force equal represent the weight of an object yields, Fg = (G m1 m2) / d2 or m1 g = (G m1 M) / d2 or g = (G M) / d2 where M is the mass of the earth Advanced Look at Weight It is easy to see that the force of gravity acts on an object when it is falling. When an object is at rest on a surface, the gravitational force still continues to act. According to the second law, only a net force would cause the motion of the object to change. Since the object remains at rest, the net force acting on it must be zero. Another force (the normal force) balances the gravitational force. This is not an example of an action/reaction pair of forces! Both forces (gravitational and normal) act on the same object. Action/reaction pairs of forces act on different objects. 2. Normal force the force exerted by a surface to support an object; symbol is FN; direction is always upward; when an object rests on a horizontal surface, the normal force equals the object’s weight; when an object is being pushed or pulled by a horizontal force, the normal force equals the object’s weight. When a push or a pull is something other than horizontal, a free-body diagram is used to determine the normal force (taking into account the vertical component of the push or the pull). A normal force is a "response" force. In other words, a surface responds to a weight resting on it by acting to oppose it to the extent necessary to "cancel out" these forces. For our purposes, normal forces only exist on a surface (if the object is in the air, there is no normal force). The normal force is not an action/reaction pair of forces with the weight. Advanced Look at the Normal Force Returning to our weight example ... An upward force (the normal force) is exerted by the table on the object. The reaction to this is the downward force exerted by the object on the table. Since these forces act on different objects, they are action/reaction forces described by Newton's Third Law. What is the reaction force acting in response to the gravitational force? The object exerts a gravitational force on the earth! Advanced Look at these forces in terms of the third law The object rests on the table and is in the Earth's gravitational field. The Earth exerts a downward gravitational force on the object equal to the object's weight. The object exerts an equal gravitational force on the Earth. The downward gravitational force exerted by the Earth and the upward gravitational force exerted by the object represent an action/reaction pair of forces. The table exerts an upward force on the object. The object exerts an equal downward force on the table. These also represent an action/reaction pair of forces. An action/reaction pair of forces never act on the same object. Action/reaction pair of forces represent the mutal interaction of two bodies. Famous Physics Puzzle: A man hitches a horse to a cart. The horse refuses to pull the cart, telling the man, "No matter how hard I pull, the cart pulls back with an equal force, and it will be impossible for me to ever pull the cart!" Answer to puzzle: • The horse exerts a force on the cart (action) and the cart exerts an equal and opposite force on the horse (reaction). The horse exerts a force on the ground (action) and the ground exerts an equal and opposite force back on the horse (reaction). There are two sets of action forces acting (for the horse). If the reaction force exerted by the ground on the horse is larger than the reaction force exerted by the sled on the horse, the horse will move forward. The cart will move forward when the force exerted by it by the horse is greater than the frictional force between the cart and the ground. Remember, there are two sets of forces acting on the cart: the force exerted by the horse on the cart (action)/the force exerted by the cart on the horse (reaction) and the force exerted by the cart on the ground (action)/the frictional force exerted by the ground on the cart (reaction). Friction • a force that usually opposes the motion of an object; symbol is Ff; direction is negative; friction is an electromagnetic force between surface atoms • Characteristics of friction: – Friction acts parallel to the surfaces in contact – Friction usually acts opposite the direction of the motion. Friction opposes the applied force. – Friction depends upon the types of surfaces in contact (All surfaces are described by a coefficient of friction, m , which is a characteristic of that surface. m has no units.) – Friction is independent of the surface area in contact.(for hard surfaces) 3. Types of friction: • Starting friction (Static Friction) opposes the beginning of motion of an object; is always greater than sliding friction • Advanced Look at Static Friction If an object is resting on a surface, there is no frictional force because there is no horizontal force. A person now pushes horizontally on the object, but not hard enough to make it move. The force of static friction exerted by the surface on the object prevents it from moving. The object does not move until you push hard enough (Fmax)to overcome static friction. – Fmax = msFN where ms is the coefficient of static friction. Sliding friction (Kinetic Friction) – opposes the motion of an object • Friction can be described mathematically: – Ff = m FN • Advanced Look at Kinetic Friction When a body is in motion along a rough surface, the force of kinetic friction acts opposite the direction of the motion. – Ff = mkFN In AP Physics, we will refer to a smooth surface as one without friction and a rough surface as one with friction. – Interactive Frictional Force Site 4. Applied force – the push or pull that "you" use to move an object; symbol is Fapp 5. Unbalanced force (or net force ) – the sum of all the forces in a direction; it is what causes the acceleration of an object SF=ma 6. Tension - a force usually associated with a rope or a cable; it is a "response" force. In other words, if one pulls on a rope, the rope "fights back" by resisting being stretched. If the rope has negligible mass, the force exerted at one end is transmitted undiminished to each adjacent piece of rope along its entire length to the other end. Note: ropes, cords, and string can only pull. They cannot push because they bend. • AP example • Two boxes, mass 10 kg and mass 20 kg, are connected by a massless string on a smooth surface. A horizontal force, Fapp pulls the 20 kg box. How would you find the acceleration of each and the tension in the cord? First, draw a free diagram for each box. Four forces are involved in the free body diagram for the 20 kg box: Fapp pulls the box horizontally to the right, the Tension (T)pulls to the left, the normal force acts up, and the gravitational force (the weight) acts down. Only the tension and the applied force act parallel to the motion. Apply Newton's second law to the 20 kg box: SF = Fapp - T = (20 kg)a. Draw a free body diagram for the 10 kg box. Three forces are involved in the free body diagram for the 10 kg box: the Tension (T)pulls to the right, the normal force acts up, and the gravitational force (the weight) acts down. (Since we said the surface was smooth, there is no friction). Only the tension acts parallel to the motion. Apply Newton's second law to the 10 kg box: SF = T = (10 kg)a. Since the boxes are connected, the two boxes have the same acceleration. We "add" the two equations and get Fapp =(30 kg)a. Once the acceleration is known, you can solve for the tension. 7. Forces involved in springs: Most mass/spring systems obey a simple relationship between force and displacement (x), known as Hooke's Law. The restoring force (F) is proportional to the displacement (x). The spring constant (k) is the proportionality constant. Felastic = -kx Forces on an Inclined Plane – The normal force exerted by the incline to support the weight, W. The parallel force is the part of the object's weight that tends to make it slide down the incline. • FN = W cos q • FP = W sin q Inclined plane problems are easier to work if one chooses the direction parallel to the incline surface as your x-axis. The y-axis is then perpendicular to the incline surface. If an object slides down an incline, three forces act on it: the normal force, the frictional force, and the gravitational force (or the object's weight). • The gravitational force can be resolved into its x and y components. Its x component is called the parallel force. Its y component is equal in magnitude and opposite in direction to the normal force exerted by the surface on the object.