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Physics B Newton’s Laws Name:_________________________ Per:____ Teacher:_____ What is a force? Newton’s First Law (Law of Inertial) What do forces cause? What is the relationship between mass and inertia? Draw a force diagram on the book. Sample Problem: A heavy block hangs from a string attached to a rod. An identical string hangs down from the bottom of the block. Which string breaks a) when the lower string is pulled with a slowly increasing force? Draw a free body diagram on the book. b) when the lower string is pulled with a quick jerk? Sample problem: Draw a force diagram and a free body diagram for a monkey hanging motionless by one arm from a vine attached to a tree. Newton’s Second Law Statement: Equation: Sample problem: Draw a force diagram and a free body diagram for a monkey hanging motionless by Units of force (SI unit only): two arms from two vines attached to neighboring trees. Working 2nd Law Problems 1. Identify the system being accelerated. 2. Define a coordinate system. 3. Identify forces by drawing a force or free body diagram. 4. Explicitly write ΣF=ma 5. Replace ΣF with the actual forces in your free body diagram. 6. Substitute numeric values, where appropriate, and solve for unknowns. 9/19/2008 1 Bertrand/Perkins Sample problem: In a grocery store, you push a Newton’s Third Law 14.5-kg cart with a force of 12.0 N. If the cart starts at rest, how far does it move in 3.00 seconds? Colloquial Definition: For every action there exists an equal and opposite reaction. Physics Definition: Sample Problem: You rest an empty glass on a table. a) Identify the forces acting on the glass with a free body diagram. Sample problem: A catcher stops a 92 mph pitch in his glove, bringing it to rest in 0.15 m. If the force exerted by the catcher is 803 N, what is the mass of the ball? b) Are these forces equal and opposite? Sample problem: A 747 jetliner lands and begins to c) Are these forces an action-reaction pair? Why or slow to a stop as it moves along the runway. Its mass why not? is 3.50 x 105 kg, its speed is 27.0 m/s, and the net braking force is 4.30 x 105 N a) What is its speed 7.50 s later? Requirements for Newton’s Laws 1. The 1st and 2nd laws require that ONE system be analyzed and that ALL the forces b) How far has it traveled in this time? on the system be accounted for. 2. The 3rd law requires that TWO systems be analyzed and that the forces of interaction between the two be accounted for. 9/19/2008 2 Bertrand/Perkins Sample problem: How long will it take a 1.0 kg block initially at rest to slide down a frictionless 20.0 m long ramp that is at a 15o angle with the horizontal? Sample Problem: A force of magnitude 7.50 N pushes three boxes as shown. Find the acceleration of the system. Sample problem: An object acted on by three forces Sample Problem: A force of magnitude 7.50 N moves with constant velocity. One force acting on the pushes three boxes as shown. Find the force that box object is in the positive x direction and has a 2 exerts on box 3. magnitude of 6.5 N; a second force has a magnitude of 4.4 N and points in the negative y direction. Find the direction and magnitude of the third force acting on the object. Sample problem: Draw a force diagram and a free body diagram for the man pushing the chair across the floor. What is the relationship between mass and inertia? Problem: A skier skis down a slope with an acceleration of 3.50 m/s2. If friction can be ignored, what is the angle of the slope with respect to the horizontal? What is the relationship between mass and weight (near the earth’s surface? 9/19/2008 3 Bertrand/Perkins Sample problem: A man weighs 150 pounds on the Sample problem: A 5-kg salmon is hanging from a surface of the earth at sea level. Calculate his fish scale in an elevator. What is the salmon’s a) mass in kg. apparent weight when the elevator is a) at rest b) weight in Newtons. b) moving downward and slowing at 3.2 m/s2? Apparent weight Apparent weight is the force which acts on a body in opposition to gravity to prevent the body from going into freefall. Sample problem: An 85-kg person is standing on a bathroom scale in an elevator. What is the person’s apparent weight a) when the elevator accelerates upward at 2.0 m/s2? Normal force Definition: The normal force is always ___________________ to a surface. b) when the elevator is moving at constant velocity between floors? Problem: derive an expression for the normal force of a box on a flat table. c) when the elevator begins to slow at the top floor at 2.0 m/s2? 9/19/2008 4 Bertrand/Perkins Problem: Derive an expression for the normal force Sample problem: A 2.5 kg book rests on a surface of a box sitting on a ramp. inclined at 28o above the horizontal Find the normal force. θ Problem: derive an expression for the normal force If the angle of the incline is reduced, do you expect an eraser being pushed up against a whiteboard by a the normal force to increase, decrease, or stay the force F. same? Explain your reasoning. Friction Definition: Problem: Derive the normal force for the box in the What causes friction? picture below. The box is sitting on the floor, but is being pulled by the force shown. Ignore friction. How is friction useful? What type of energy does friction often generate? What other force is directly related to the friction force? Why do you think this force causes friction to F = 20 N exist? 40o 6.0 kg 9/19/2008 5 Bertrand/Perkins What are the two main types of friction? Sample problem: A 10-kg wooden box rests on a wooden floor. The coefficient of static friction is 0.50, and the coefficient of kinetic friction is 0.30. What is the friction force between the box and floor if Static Friction a) no force horizontal force is applied to the box? Defition: Equation: b) a 20 N horizontal force is applied to the box? Kinetic Friction Defition: Equation: c) a 60 N horizontal force is applied to the box? Problem: A 10-kg box rests on a ramp that is laying flat. The coefficient of static friction is 0.50, and the coefficient of kinetic friction is 0.30. a) What is the maximum horizontal force that can be applied to the box before it begins to slide? Problem: A 10-kg wooden box rests on a wooden ramp. The coefficient of static friction is 0.50, and the coefficient of kinetic friction is 0.30. What is the friction force between the box and ramp if a) the ramp is at a 25o angle? b) What force is necessary to keep the box sliding at constant velocity? 9/19/2008 6 Bertrand/Perkins b) the ramp is at a 45o angle? Springs Hooke’s Law Definition: Equation: Why is the sign in the Hooke’s Law equation c) what is the acceleration of the box when the ramp negative? is at 45o? Problem: A 1.50 kg object hangs motionless from a spring with a force constant of k = 250 N/m. How far is the spring stretched from its equilibrium length? Tension Define tension: Sample problem: A 1,500 kg crate hangs from a crane cable. a) What is the tension in the cable when the crate is Problem: A 1.80 kg object is connected to a spring motionless? Ignore the mass of the cable. of force constant 120 N/m. How far is the spring stretched if it is used to drag the object across a floor at constant velocity? Assume the coefficient of kinetic friction is 0.60. b) Suppose the crane accelerates the crate upward at 1.2 m/s2. What is the tension in the cable now? 9/19/2008 7 Bertrand/Perkins Problem: A 5.0 kg object is connected to a 10.0 kg Sample problem: A 10 kg block rests on a table object by a string. If a pulling force F of 20 N is connected by a string to a 5 kg block Find the applied to the 5.0 kg object, minimum coefficient of static friction for which the a) what is the acceleration of the system? blocks remain stationary. m1 Frictionless table m2 b)what is the tension in the string connecting the objects? Problem: Derive the acceleration, assuming a coefficient of friction of 0.20 between the table and the 3.0 kg block. Determine the tension in the string. Pulleys What is the main effect of a magic pulley on a 3.0 kg 5.0 kg Newton’s 2nd Law problem? 35o Sample problem: Derive an expression for the acceleration due to gravity of the system below, and for the tension in the string. m1 Frictionless table m2 Uniform Circular Motion Definition: Question: Why is uniform circular motion accelerated motion? 9/19/2008 8 Bertrand/Perkins Problem: A 1200-kg car rounds a corner of radius r Question: What is centrifugal force? = 45 m. If the coefficient of static friction between tires and the road is 0.93, what is the maximum velocity the car can have without skidding? Question: If your body feels flung in a certain direction, in which direction is the net force acting upon your body. Centripetal Acceleration Definition: Problem: You whirl a 1.0 kg stone in a horizontal circle about your head. The rope attached to the stone is 1.5 m long. What is the tension in the rope? (The rope makes a Picture: 10o angle with the horizontal). Equation: b) How fast is the stone moving? Centripetal Force Definition: Newton’s 2nd Law with centripetal acceleration: Forces that can cause centripetal acceleration: 9/19/2008 9 Bertrand/Perkins Newton’s Law of Universal Gravitation planet Jupiter does”. Is Joe correct? (Assume a 100-lb Definition: Lab 1.0 meter away, and Jupiter at its farthest distance from Earth). Equation Problem: a) How much force does the earth exert on the moon? Problem: What would be your weight if you were orbiting the earth in a satellite at an altitude of 3,000,000 km above the earth’s surface? (Note that b) How much force does the moon exert on the earth? even though you are apparently weightless, gravity is still exerting a force on your body, and this is your actual weight.) Problem: Using centripetal force and Newton’s Law of Universal Gravitation, derive the mass of the sun using the orbit of the earth. Problem: Derive the numeric value of the acceleration due to gravity on the surface of the earth. Start with Newton’s 2nd Law. Problem: Sally, an astrology buff, claims that the position of the planet Jupiter influences events in her life. She surmises this is due to its gravitational pull. Joe scoffs at Sally and says “your Labrador Retriever exerts more gravitational pull on your body than the 9/19/2008 10 Bertrand/Perkins Problem: What is the acceleration due to gravity at Problem: What velocity does a satellite in orbit an altitude equal to twice the earth’s radius? about the earth at an altitude of 25,000 km have? What is the period of this satellite? Problem: What is the acceleration due to gravity at the surface of the moon? Problem: A geosynchronous satellite is one which remains above the same point on the earth. Such a satellite orbits the earth in 24 hours, thus matching the earth's rotation. How high must must a geosynchronous satellite be above the surface to maintain a geosynchronous orbit? Kepler’s Laws 1. Planets orbit the sun in elliptical orbits, with the sun at a focus. 2. Planets orbiting the sun carve out equal area triangles in equal times. 3. The planet’s year is related to its distance from the sun in a predictable way. Problem: Using Newton’s Law of Universal Gravitation, derive a formula to show how the period of a planet’s orbit varies with the radius of that orbit. Assume a nearly circular orbit. (This is a derivation of Kepler’s 3rd Law.) 9/19/2008 11 Bertrand/Perkins Physics B Newton’s Laws Name:_________________________ Per:____ Teacher:_____ A-176 B1. In the system shown at right, the block of mass M1 is on a rough horizontal table. The string that attaches it to the block of mass M2 passes over a frictionless pulley of negligible mass. The coefficient of kinetic friction µk between M1 and the table is less than the coefficient of static friction µs a. On the diagram below, draw and identify all the forces acting on the block of mass M1. M1 b. In terms of M1 and M2 determine the minimum value of µs that will prevent the blocks from moving. The blocks are set in motion by giving M2 a momentary downward push. In terms of M1, M2, µk, and g, determine each of the following: c. The magnitude of the acceleration of M1 d. The tension in the string. 9/19/2008 12 Bertrand/Perkins A-178 B1. An object of mass M on a string is whirled with increasing speed in a horizontal circle, as shown at right. When the string breaks, the object has speed vo and the circular path has radius R and is a height h above the ground. Neglect air friction. a. Determine the following, expressing all answers in terms of h, vo, and g. i. The time required for the object to hit the ground after the string breaks ii. The horizontal distance the object travels from the time the string breaks until it hits the ground iii. The speed of the object just before it hits the ground b. On the figure below, draw and label all the forces acting on the object when it is in the position shown in the diagram above. . c. Determine the tension in the string just before the string breaks. Express your answer in terms of M, R, vo, and g. 9/19/2008 13 Bertrand/Perkins A-182 B1. A student whose normal weight is 500 newtons stands on a scale in an elevator and records the scale reading as a function of time. The data are shown in the graph above. At time t = 0, the elevator is at displacement x = 0 with velocity v = 0. Assume that the positive directions for displacement, velocity, and acceleration are upward. a. On the diagram below, draw and label all of the forces on the student at t = 8 seconds. b. Calculate the acceleration a of the elevator for each 5-second interval. i. Indicate your results by completing the following table. Time Interval (s) 0-5 5-10 10-15 15-20 a (m|s2) ii. Plot the acceleration as a function of time on the following graph. 9/19/2008 14 Bertrand/Perkins c. Determine the velocity v of the elevator at the end of each 5-second interval. i. Indicate your results by completing the following table. Time (s) 0-5 5-10 10-15 15-20 v (m| s) ii. Plot the velocity as a function of time on the following graph. d. Determine the displacement x of the elevator above the starting point at the end of each 5-second interval. i. Indicate your results by completing the following table. Time (s) 0-5 5-10 10-15 15-20 x (m) ii. Plot the displacement as a function of time on the following graph. 9/19/2008 15 Bertrand/Perkins S-099 B5 (10 pts) A coin C of mass 0.0050 kg is placed on a horizontal disk at a distance of 0.14 m from the center, as shown above. The disk rotates at a constant rate in a counterclockwise direction as seen from above. The coin does not slip, and the time it takes for the coin to make a complete revolution is 1.5 s. a. The figure below shows the disk and coin as viewed from above. Draw and label vectors on the figure below to show the instantaneous acceleration and linear velocity vectors for the coin when it is at the position shown. b. Determine the linear speed of the coin. c. The rate of rotation of the disk is gradually increased. The coefficient of static friction between the coin and the disk is 0.50. Determine the linear speed of the coin when it just begins to slip. d. If the experiment in part (c) were repeated with a second, identical coin glued to the top of the first coin, how would this affect the answer to part (c) ? Explain your reasoning. 9/19/2008 16 Bertrand/Perkins