AP* Newton s Laws Free Response Questions

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1 1981 Q1 AP* Newton s Laws Free Response Questions A 10-kilogram block is pushed along a rough horizontal surface by a constant horizontal force F as shown above. At time t = 0, the velocity v of the block is 6.0 meters per second in the same direction as the force. The coefficient of sliding friction is 0.2. Assume g = 10 meters per second squared. (a) Calculate the force F necessary to keep the velocity constant. The force is now changed to a larger constant value F. The block accelerates so that its kinetic energy increases by 60 joules while it slides a distance of 4.0 meters. (b) Calculate the force F. (c) Calculate the acceleration of the block. (1) AP is a registered trademark of the College Board. The College Board was not involved in the production of and does not endorse this product. (2) Test Questions are Copyright by College Entrance Examination Board, Princeton, NJ. All rights reserved. For face-to-face teaching purposes, classroom teachers are permitted to reproduce the questions. Web or Mass distribution prohibited.

2 AP* Newton s Laws Free Response Questions page Q1 A box of uniform density weighing 100 newtons moves in a straight line with constant speed along a horizontal surface. The coefficient of sliding friction is 0.4 and a rope exerts a force F in the direction of motion as shown above. (a) On the diagram below, draw and identify all the forces on the box. (b) Calculate the force F exerted by the rope that keeps the box moving with constant speed.

3 AP* Newton s Laws Free Response Questions page 3 (c) A horizontal force F', applied at a height 5/3 meters above the surface as shown in the diagram above, is just sufficient to cause the box to begin to tip forward about an axis through point P. The box is 1 meter wide and 2 meters high. Calculate the force F.

4 AP* Newton s Laws Free Response Questions page Q1 In the system shown above, the block of mass M 1 is on a rough horizontal table. The string that attaches it to the block of mass M 2 passes over a frictionless pulley of negligible mass. The coefficient of kinetic friction μ k between M 1 and the table is less than the coefficient of static friction μ s (a) On the diagram below, draw and identify all the forces acting on the block of mass M 1. M 1 (b) In terms of M 1 and M 2 determine the minimum value of μ s that will prevent the blocks from moving.

5 AP* Newton s Laws Free Response Questions page 5 The blocks are set in motion by giving M 2 a momentary downward push. In terms of M 1, M 2, μ k, and g, determine each of the following: (c) The magnitude of the acceleration of M 1 (d) The tension in the string.

6 2006 Q1 (15 points) AP* Newton s Laws Free Response Questions page 6 An ideal spring of unstretched length 0.20 m is placed horizontally on a frictionless table as shown above. One end of the spring is fixed and the other end is attached to a block of mass M = 8.0 kg. The 8.0 kg block is also attached to a massless string that passes over a small frictionless pulley. A block of mass m = 4.0 kg hangs from the other end of the string. When this spring-and-blocks system is in equilibrium, the length of the spring is 0.25 m and the 4.0 kg block is 0.70 m above the floor. (a) On the figures below, draw free-body diagrams showing and labeling the forces on each block when the system is in equilibrium. M = 8.0 kg m = 4.0 kg (b) Calculate the tension in the string. (c) Calculate the force constant of the spring.

7 AP* Newton s Laws Free Response Questions page 7 The string is now cut at point P. (d) Calculate the time taken by the 4.0 kg block to hit the floor. (e) Calculate the frequency of oscillation of the 8.0 kg block. (f) Calculate the maximum speed attained by the 8.0 kg block.

8 2007 Q1 (15 points) AP* Newton s Laws Free Response Questions page 8 An empty sled of mass 25 kg slides down a muddy hill with a constant speed of 2.4 m/s. The slope of the hill is inclined at an angle of 15 with the horizontal as shown in the figure above. (a) Calculate the time it takes the sled to go 21 m down the slope. (b) On the dot below that represents the sled, draw and label a free-body diagram for the sled as it slides down the slope.

9 AP* Newton s Laws Free Response Questions page 9 (c) Calculate the frictional force on the sled as it slides down the slope. (d) Calculate the coefficient of friction between the sled and the muddy surface of the slope. (e) The sled reaches the bottom of the slope and continues on the horizontal ground. Assume the same coefficient of friction. i. In terms of velocity and acceleration, describe the motion of the sled as it travels on the horizontal ground.

10 AP* Newton s Laws Free Response Questions page 10 ii. On the axes below, sketch a graph of speed u versus time t for the sled. Include both the sled s travel down the slope and across the horizontal ground. Clearly indicate with the symbol t l the time at which the sled leaves the slope.

11 2003B Q1 (15 points) AP* Newton s Laws Free Response Questions page 11 An airplane accelerates uniformly from rest. A physicist passenger holds up a thin string of negligible mass to which she has tied her ring, which has a mass m. She notices that as the plane accelerates down the runway, the string makes an angle θ with the vertical as shown above. (a) In the space below, draw a free-body diagram of the ring, showing and labeling all the forces present. The plane reaches a takeoff speed of 65 m/s after accelerating for a total of 30 s. (b) Determine the minimum length of the runway needed.

12 AP* Newton s Laws Free Response Questions page 12 (c) Determine the angle θ that the string makes with the vertical during the acceleration of the plane before it leaves the ground. (d) What additional information would be needed in order to estimate the mechanical energy of the airplane at the instant of takeoff? Explain your answer.

13 2005 Q2 (10 points) AP* Newton s Laws Free Response Questions page 13 A simple pendulum consists of a bob of mass 1.8 kg attached to a string of length 2.3 m. The pendulum is held at an angle of 30 from the vertical by a light horizontal string attached to a wall, as shown above. (a) On the figure below, draw a free-body diagram showing and labeling the forces on the bob in the position shown above.

14 AP* Newton s Laws Free Response Questions page 14 (b) Calculate the tension in the horizontal string. (c) The horizontal string is now cut close to the bob, and the pendulum swings down. Calculate the speed of the bob at its lowest position.

15 2005B Q1 (10 points) AP* Newton s Laws Free Response Questions page 15 A student of mass m stands on a platform scale in an elevator in a tall building. The positive direction for all vector quantities is upward. (a) Draw a free-body diagram showing and labeling all the forces acting on the student, who is represented by the dot below. (b) Derive an expression for the reading on the scale in terms of the acceleration a of the elevator, the mass m of the student, and fundamental constants.

16 AP* Newton s Laws Free Response Questions page 16 An inspector provides the student with the following graph showing the acceleration a of the elevator as a function of time t. (c) i. During what time interval(s) is the force exerted by the platform scale on the student a maximum value? ii. Calculate the magnitude of that maximum force for a 45 kg student. (d) During what time interval(s) is the speed of the elevator constant?

17 AP* Newton s Laws Free Response Questions page Q1 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) a (m s 2 ) ii. Plot the acceleration as a function of time on the following graph.

18 AP* Newton s Laws Free Response Questions page 18 (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) 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.

19 AP* Newton s Laws Free Response Questions page 19 i. Indicate your results by completing the following table. Time (s) x (m) ii. Plot the displacement as a function of time on the following graph.

20 2002 Q1 (15 points) AP* Newton s Laws Free Response Questions page 20 A model rocket of mass kg is launched vertically with an engine that is ignited at time t = 0, as shown above. The engine provides an impulse of 20.0 N s by firing for 2.0 s. Upon reaching its maximum height, the rocket deploys a parachute, and then descends vertically to the ground. (a) On the figures below, draw and label a free-body diagram for the rocket during each of the following intervals. i. While the engine is ii. After the engine stops, iii. After the parachute is firing but before the parachute is deployed (b) Determine the magnitude of the average acceleration of the rocket during the 2 s firing of the engine.

21 AP* Newton s Laws Free Response Questions page 21 (c) What maximum height will the rocket reach? (d) At what time after t = 0 will the maximum height be reached?

22 AP* Newton s Laws Free Response Questions page Q1 Three blocks of masses 1.0, 2.0, and 4.0 kilograms are connected by massless strings, one of which passes over a frictionless pulley of negligible mass, as shown above. Calculate each of the following. (a) The acceleration of the 4-kilogram block (b) The tension in the string supporting the 4-kilogram block (c) The tension in the string connected to the l-kilogram block

23 AP* Newton s Laws Free Response Questions page Q2 A crane is used to hoist a load of mass m 1 = 500 kilograms. The load is suspended by a cable from a hook of mass m 2 = 50 kilograms, as shown in the diagram above. The load is lifted upward at a constant acceleration of 2 m/s 2. (a) On the diagrams below draw and label the forces acting on the hook and the forces acting on the load as they accelerate upward (b) Determine the tension T 1 in the lower cable and the tension T 2 in the upper cable as the hook and load are accelerated upward at 2 m/s 2. Use g = 10 m/s².

24 2003 Q1 (15 points) AP* Newton s Laws Free Response Questions page 24 A rope of negligible mass passes over a pulley of negligible mass attached to the ceiling, as shown above. One end of the rope is held by Student A of mass 70 kg, who is at rest on the floor. The opposite end of the rope is held by Student B of mass 60 kg, who is suspended at rest above the floor. (a) On the dots below that represent the students, draw and label free-body diagrams showing the forces on Student A and on Student B. B A (b) Calculate the magnitude of the force exerted by the floor on Student A.

25 AP* Newton s Laws Free Response Questions page 25 Student B now climbs up the rope at a constant acceleration of 0.25 m/s2 with respect to the floor. (c) Calculate the tension in the rope while Student B is accelerating. (d) As Student B is accelerating, is Student A pulled upward off the floor? Justify your answer. (e) With what minimum acceleration must Student B climb up the rope to lift Student A upward off the floor?

26 AP* Newton s Laws Free Response Questions page Q1 A 5.0-kilogram monkey hangs initially at rest from two vines, A and B. as shown above. Each of the vines has length 10 meters and negligible mass. (a) On the figure below, draw and label all of the forces acting on the monkey. (Do not resolve the forces into components, but do indicate their directions.) (b) Determine the tension in vine B while the monkey is at rest.

27 AP* Newton s Laws Free Response Questions page 27 The monkey releases vine A and swings on vine B. Neglect air resistance. (c) Determine the speed of the monkey as it passes through the lowest point of its first swing. (d) Determine the tension in vine B as the monkey passes through the lowest point of its first swing.

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