AP Physics Free Response Practice Oscillations 1975B7. A pendulum consists of a small object of mass m fastened to the end of an inextensible cord of length L. Initially, the pendulum is drawn aside through an angle of 60 with the vertical and held by a horizontal string as shown in the diagram above. This string is burned so that the pendulum is released to swing to and fro. a. In the space below draw a force diagram identifying all of the forces acting on the object while it is held by the string. b. Determine the tension in the cord before the string is burned. c. Show that the cord, strong enough to support the object before the string is burned, is also strong enough to support the object as it passes through the bottom of its swing. d. The motion of the pendulum after the string is burned is periodic. Is it also simple harmonic? Why, or why not? 1983B2. A block of mass M is resting on a horizontal, frictionless table and is attached as shown above to a relaxed spring of spring constant k. A second block of mass 2M and initial speed v o collides with and sticks to the first block Develop expressions for the following quantities in terms of M, k, and v o a. v, the speed of the blocks immediately after impact b. x, the maximum distance the spring is compressed c. T, the period of the subsequent simple harmonic motion 301
1995B1. As shown above, a 0.20-kilogram mass is sliding on a horizontal, frictionless air track with a speed of 3.0 meters per second when it instantaneously hits and sticks to a 1.3-kilogram mass initially at rest on the track. The 1.3-kilogram mass is connected to one end of a massless spring, which has a spring constant of 100 newtons per meter. The other end of the spring is fixed. a. Determine the following for the 0.20-kilogram mass immediately before the impact. i. Its linear momentum ii. Its kinetic energy b. Determine the following for the combined masses immediately after the impact. i. The linear momentum ii. The kinetic energy After the collision, the two masses undergo simple harmonic motion about their position at impact. c. Determine the amplitude of the harmonic motion. d. Determine the period of the harmonic motion. 1996B2. A spring that can be assumed to be ideal hangs from a stand, as shown above. a. You wish to determine experimentally the spring constant k of the spring. i. What additional, commonly available equipment would you need? ii. What measurements would you make? iii. How would k be determined from these measurements? b. Assume that the spring constant is determined to be 500 N/m. A 2.0-kg mass is attached to the lower end of the spring and released from rest. Determine the frequency of oscillation of the mass. c. Suppose that the spring is now used in a spring scale that is limited to a maximum value of 25 N, but you would like to weigh an object of mass M that weighs more than 25 N. You must use commonly available equipment and the spring scale to determine the weight of the object without breaking the scale. i. Draw a clear diagram that shows one way that the equipment you choose could be used with the spring scale to determine the weight of the object, ii. Explain how you would make the determination. 302
2005B2 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. (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. (d) How long will it take the bob to reach the lowest position for the first time? 303
2005B2B A simple pendulum consists of a bob of mass 0.085 kg attached to a string of length 1.5 m. The pendulum is raised to point Q, which is 0.08 m above its lowest position, and released so that it oscillates with small P and Q as shown below. (a) On the figures below, draw free-body diagrams showing and labeling the forces acting on the bob in each of the situations described. i. When it is at point P ii. When it is in motion at its lowest position (b) Calculate the speed v of the bob at its lowest position. (c) Calculate the tension in the string when the bob is passing through its lowest position. (d) Describe one modification that could be made to double the period of oscillation. 304
2006B1 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. 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. C1989M3. A 2-kilogram block is dropped from a height of 0.45 meter above an uncompressed spring, as shown above. The spring has an elastic constant of 200 newtons per meter and negligible mass. The block strikes the end of the spring and sticks to it. a. Determine the speed of the block at the instant it hits the end of the spring b. Determine the force in the spring when the block reaches the equilibrium position c. Determine the distance that the spring is compressed at the equilibrium position d. Determine the speed of the block at the equilibrium position e. Determine the resulting amplitude of the oscillation that ensues f. Is the speed of the block a maximum at the equilibrium position, explain. g. Determine the period of the simple harmonic motion that ensues 305
1990M3. A 5-kilogram block is fastened to an ideal vertical spring that has an unknown spring constant. A 3-kilogram block rests on top of the 5-kilogram block, as shown above. a. When the blocks are at rest, the spring is compressed to its equilibrium position a distance of 1 = 20 cm, from its original length. Determine the spring constant of the spring The 3 kg block is then raised 50 cm above the 5 kg block and dropped onto it. b. Determine the speed of the combined blocks after the collision c. Setup, plug in known values, but do not solve an equation to determine the amplitude 2 of the resulting oscillation d. Determine the resulting frequency of this oscillation. e. Where will the block attain its maximum speed, explain. f. Is this motion simple harmonic. 306
(2000 M1) You are conducting an experiment to measure the acceleration due to gravity g u at an unknown location. In the measurement apparatus, a simple pendulum swings past a photogate located at the pendulum's lowest point, which records the time t 10 for the pendulum to undergo 10 full oscillations. The pendulum consists of a sphere of mass m at the end of a string and has a length l. There are four versions of this apparatus, each with a different length. All four are at the unknown location, and the data shown below are sent to you during the experiment. (cm) t 10 (s) 12 7.62 18 8.89 21 10.09 32 12.08 T (s) 2 T (s 2 ) a. For each pendulum, calculate the period T and the square of the period. Use a reasonable number of significant figures. Enter these results in the table above. b. On the axes below, plot the square of the period versus the length of the pendulum. Draw a best-fit straight line for this data. c. Assuming that each pendulum undergoes small amplitude oscillations, from your fit, determine the experimental value g exp of the acceleration due to gravity at this unknown location. Justify your answer. d. If the measurement apparatus allows a determination of gu that is accurate to within 4%, is your experimental value in agreement with the value 9.80 m/s 2? Justify your answer. e. Someone informs you that the experimental apparatus is in fact near Earth's surface, but that the experiment has been conducted inside an elevator with a constant acceleration a. If the elevator is moving down, determine the direction of the elevator's acceleration, justify your answer. 307
C2003M2. An ideal massless spring is hung from the ceiling and a pan suspension of total mass M is suspended from the end of the spring. A piece of clay, also of mass M, is then dropped from a height H onto the pan and sticks to it. Express all algebraic answers in terms of the given quantities and fundamental constants. (a) Determine the speed of the clay at the instant it hits the pan. (b) Determine the speed of the clay and pan just after the clay strikes it. (c) After the collision, the apparatus comes to rest at a distance H/2 below the current position. Determine the spring constant of the attached spring. (d) Determine the resulting period of oscillation 308
C2008M3 In an experiment to determine the spring constant of an elastic cord of length 0.60 m, a student hangs the cord from a rod as represented above and then attaches a variety of weights to the cord. For each weight, the student allows the weight to hang in equilibrium and then measures the entire length of the cord. The data are recorded in the table below: (a) Use the data to plot a graph of weight versus length on the axes below. Sketch a best-fit straight line through the data. (b) Use the best-fit line you sketched in part (a) to determine an experimental value for the spring constant k of the cord. The student now attaches an object of unknown mass m to the cord and holds the object adjacent to the point at which the top of the cord is tied to the rod, as shown. When the object is released from rest, it falls 1.5 m before stopping and turning around. Assume that air resistance is negligible. (c) Calculate the value of the unknown mass m of the object. (d) i. Determine the magnitude of the force in the cord when the when the mass reaches the equilibrium position. ii. Determine the amount the cord has stretched when the mass reaches the equilibrium position. iii. Calculate the speed of the object at the equilibrium position iv. Is the speed in part iii above the maximum speed, explain your answer. 309
Supplemental One end of a spring of spring constant k is attached to a wall, and the other end is attached to a block of mass M, as shown above. The block is pulled to the right, stretching the spring from its equilibrium position, and is then held in place by a taut cord, the other end of which is attached to the opposite wall. The spring and the cord have negligible mass, and the tension in the cord is F T. Friction between the block and the surface is negligible. Express all algebraic answers in terms of M, k, F T, and fundamental constants. (a) On the dot below that represents the block, draw and label a free-body diagram for the block. (b) Calculate the distance that the spring has been stretched from its equilibrium position. The cord suddenly breaks so that the block initially moves to the left and then oscillates back and forth. (c) Calculate the speed of the block when it has moved half the distance from its release point to its equilibrium position. (d) Calculate the time after the cord breaks until the block first reaches its position furthest to the left. (e) Suppose instead that friction is not negligible and that the coefficient of kinetic friction between the block and the surface is µ k. After the cord breaks, the block again initially moves to the left. Calculate the initial acceleration of the block just after the cord breaks. 310