1992 Q1 AP* Circular & Gravitation Free Response Questions A 0.10-kilogram solid rubber ball is attached to the end of a 0.80-meter length of light thread. The ball is swung in a vertical circle, as shown in the diagram above. Point P, the lowest point of the circle, is 0.20 meter above the floor. The speed of the ball at the top of the circle is 6.0 meters per second, and the total energy of the ball is kept constant. (a) Determine the total energy of the ball, using the floor as the zero point for gravitational potential energy. (b) Determine the speed of the ball at point P, the lowest point of the circle. (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 1984-2008 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.
AP* Circular & Gravitation Free Response Questions page 2 (c) Determine the tension in the thread at i. the top of the circle; ii. the bottom of the circle. The ball only reaches the top of the circle once before the thread breaks when the ball is at the lowest point of the circle. (d) Determine the horizontal distance that the ball travels before hitting the floor.
AP* Circular & Gravitation Free Response Questions page 3 1978 Q1 A 0.5-kilogram object rotates freely in a vertical circle at the end of a string of length 2 meters as shown above. As the object passes through point P at the top of the circular path, the tension in the string is 20 newtons. Assume g = 10 meters per second squared. (a) On the following diagram of the object, draw and clearly label all significant forces on the object when it is at point P. (b) Calculate the speed of the object at point P.
AP* Circular & Gravitation Free Response Questions page 4 (c) Calculate the increase in Kinetic energy of the object as it moves from point P to point Q. (d) Calculate the tension in the string as the object passes through point Q.
AP* Circular & Gravitation Free Response Questions page 5 1984 Q1 A ball of mass M attached to a string of length L moves in a circle in a vertical plane as shown above. At the top of the circular path, the tension in the string is twice the weight of the ball. At the bottom, the ball just clears the ground. Air resistance is negligible. Express all answers in terms of M, L, and g. (a) Determine the magnitude and direction of the net force on the ball when it is at the top. (b) Determine the speed v 0 of the ball at the top.
AP* Circular & Gravitation Free Response Questions page 6 The string is then cut when the ball is at the top. (c) Determine the time it takes the ball to reach the ground. (d) Determine the horizontal distance the ball travels before hitting the ground.
AP* Circular & Gravitation Free Response Questions page 7 1977 Q2 A box of mass M, held in place by friction, rides on the flatbed of a truck which is traveling with constant speed v. The truck is on an unbanked circular roadway having radius of curvature R. a. On the diagram provided above, indicate and clearly label all the force vectors acting on the box. b. Find what condition must be satisfied by the coefficient of static friction μ between the box and the truck bed. Express your answer in terms of v, R, and g. If the roadway is properly banked, the box will still remain in place on the truck for the same speed v even when the truck bed is frictionless. c. On the diagram below indicate and clearly label the two forces acting on the box under these conditions d. Which, if either, of the two forces acting on the box is greater in magnitude?
AP* Circular & Gravitation Free Response Questions page 8 2002B Q2 (15 points) A ball attached to a string of length swings in a horizontal circle, as shown above, with a constant speed. The string makes an angle θ with the vertical, and T is the magnitude of the tension in the string. Express your answers to the following in terms of the given quantities and fundamental constants. (a) On the figure below, draw and label vectors to represent all the forces acting on the ball when it is at the position shown in the diagram. The lengths of the vectors should be consistent with the relative magnitudes of the forces. (b) Determine the mass of the ball.
AP* Circular & Gravitation Free Response Questions page 9 (c) Determine the speed of the ball. (d) Determine the frequency of revolution of the ball. (e) Suppose that the string breaks as the ball swings in its circular path. Qualitatively describe the trajectory of the ball after the string breaks but before it hits the ground.
2009B Q1 (15 points) AP* Circular & Gravitation Free Response Questions page 10 An experiment is performed using the apparatus above. A small disk of mass m1 on a frictionless table is attached to one end of a string. The string passes through a hole in the table and an attached narrow, vertical plastic tube. An object of mass m2 is hung at the other end of the string. A student holding the tube makes the disk rotate in a circle of constant radius r, while another student measures the period P. (a) Derive the equation 2 that relates P and m 2. The procedure is repeated, and the period P is determined for four different values of m 2, where m1 = 0.012 kg and r = 0.80 m. The data, which are presented below, can be used to compute an experimental value for g.
AP* Circular & Gravitation Free Response Questions page 11 (b) What quantities should be graphed to yield a straight line with a slope that could be used to determine g? (c) On the grid below, plot the quantities determined in part (b), label the axes, and draw the best-fit line to the data. You may use the blank rows above to record any values you may need to calculate. (d) Use your graph to calculate the experimental value of g.
AP* Circular & Gravitation Free Response Questions page 12 1994 Q2 A track consists of a frictionless arc XY, which is a quarter-circle of radius R, and a rough horizontal section YZ. Block A of mass M is released from rest at point X, slides down the curved section of the track, and collides instantaneously and inelastically with identical block B at point Y. The two blocks move together to the right, sliding past point P, which is a distance from point Y. The coefficient of kinetic friction between the blocks and the horizontal part of the track is μ Express your answers in terms of M,, μ, R, and g. (a) Determine the speed of block A just before it hits block B. (b) Determine the speed of the combined blocks immediately after the collision.
AP* Circular & Gravitation Free Response Questions page 13 (c) Determine the amount of kinetic energy lost due to the collision. (d) The specific heat of the material used to make the blocks is c. Determine the temperature rise that results from the collision in terms of c and the other given quantities. (Assume that no energy is transferred to the track or to the air surrounding the blocks.) (e) Determine the additional thermal energy that is generated as the blocks move from Y to P
1995 Q3 (15 points) AP* Circular & Gravitation Free Response Questions page 14 Part of the track of an amusement park roller coaster is shaped as shown above. A safety bar is oriented lengthwise along the top of each car. In one roller coaster car, a small 0.10-kilogram ball is suspended from this bar by a short length of light, inextensible string. (a) Initially, the car is at rest at point A. i. On the diagram below, draw and label all the forces acting on the 0.10-kilogram ball. ii. Calculate the tension in the string. The car is then accelerated horizontally, goes up a 30 incline, goes down a 30 incline, and then goes around a vertical circular loop of radius 25 meters. For each of the four situations described in parts (b) to (e), do all three of the following. In each situation, assume that the ball has stopped swinging back and forth. 1) Determine the horizontal component T h of the tension in the string in newtons and record your answer in the space provided. 2) Determine the vertical component T v of the tension in the string in newtons and record your answer in the space provided. 3) Show on the adjacent diagram the approximate direction of the string with respect to the vertical. The dashed line shows the vertical in each situation. (b) The car is at point B moving horizontally to the right with an acceleration of 5.0 m/s. T h = T v =
AP* Circular & Gravitation Free Response Questions page 15 (c) The car is at point C and is being pulled up the 30 incline with a constant speed of 30 m/s. T h = T v = (d) The car is at point D moving down the 30 incline with an acceleration of 5.0 m/s 2. T h = T v = (e) The car is at point E moving upside down with an instantaneous speed of 25 m/s and no tangential acceleration at the top of the vertical loop of radius 25 meters. T h = T v =
2004 Q1 (15 points) AP* Circular & Gravitation Free Response Questions page 16 A designer is working on a new roller coaster, and she begins by making a scale model. On this model, a car of total mass 0.50 kg moves with negligible friction along the track shown in the figure above. The car is given an initial speed v 0 = 1.5 m s at the top of the first hill of height 2.0 m. Point A is located at a height of 1.9 m at the top of the second hill, the upper part of which is a circular arc of radius 0.95 m. (a) Calculate the speed of the car at point A. (b) On the figure of the car below, draw and label vectors to represent the forces on the car at point A.
AP* Circular & Gravitation Free Response Questions page 17 (c) Calculate the magnitude of the force of the track on the car at point A. (d) In order to stop the car at point A, some friction must be introduced. Calculate the work that must be done by the friction force in order to stop the car at point A. (e) Explain how to modify the track design to cause the car to lose contact with the track at point A before descending down the track. Justify your answer.