AP Physics Problems Simple Harmonic Motion, Mechanical Waves and Sound 1. 1977-5 (Mechanical Waves/Sound) Two loudspeakers, S 1 and S 2 a distance d apart as shown in the diagram below left, vibrate in phase and emit sound waves of equal amplitude and wavelength λ. a. Describe how sound intensity I varies as a function of position x along the line segment OA. On your own paper, using axes like those above center, sketch the graph of this function. b. Assume d << L. On your own paper, using axes like those above right, sketch a graph of the sound intensity I as a function of position y along the y axis. c. Assume that d = 2.0 meters and that the speed of sound is 360 meters per second. Find the lowest speaker frequency which will yield the minimum sound intensity along the line BB'. d. 2. 1983-2 (Simple Harmonic Motion, Momentum) 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 3. 1995-1 (Simple Harmonic Motion) As shown to the right, a 0.20 kg mass is sliding on a horizontal, frictionless air track with a speed of 3.0 m/s when it instantaneously hits and sticks to a 1.3 kg mass initially at rest on the track. The 1.3 kg 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 kg 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.
4. 1995-6 (Mechanical Waves) A hollow tube of length l. open at both ends as shown above, is held in midair. A tuning fork with a frequency f o vibrates at one end of the tube and causes the air in the tube to vibrate at its fundamental frequency. Express your answers in terms of l and f o. a. Determine the wavelength of the sound. b. Determine the speed of sound in the air inside the tube. c. Determine the next higher frequency at which this air column would resonate. The tube is submerged in a large, graduated cylinder filled with water. The tube is slowly raised out of the water and the same tuning fork, vibrating with frequency f O, is held a fixed distance from the top of the tube. d. Determine the height h of the tube above the water when the air column resonates for the first time. Express your answer in terms of l. 5. 1997-3 (Simple Harmonic Motion) A rigid rod of mass m and length l is suspended from two identical springs of negligible mass as shown in the diagram to the right. The upper ends of the springs are fixed in place and the springs stretch a distance d under the weight of the suspended rod. a. Determine the spring constant k of each spring in terms of the other given quantities and fundamental constants. As shown to the right, the upper end of the springs are connected by a circuit branch containing a battery of emf ε and a switch S so that a complete circuit is formed with the metal rod and springs. The circuit has a total resistance R, represented by the resistor in the diagram. The rod is in a uniform magnetic field directed perpendicular to the page. The upper ends of the springs remain fixed in place and the switch S is closed. When the system comes to equilibrium, the rod is lowered an additional distance Δd. b. What is the direction of the magnetic field relative to the coordinate axes shown to the right? c. Determine the magnitude of the magnetic field in terms of m, Q, d, Δd, ε, R, and fundamental constants. d. When the switch is suddenly opened, the rod oscillates. For these oscillations, determine the following quantities in terms of d, Δd, and fundamental constants: i. The period ii. The maximum speed of the rod
6. 1998-5 (Mechanical Waves) To demonstrate standing waves, one end of a string is attached to a tuning fork with frequency of 120 Hz. The other end of the string passes over a pulley and is connected to a suspended mass M as shown in the figure above. The value of M is such that the standing wave pattern has four "loops." The length of the string from the tuning fork to the point where the string touches the top of the pulley is 1.20 m. The linear density of the string is 1.0 10-4 kg/m, and remains constant throughout the experiment. a. Determine the wavelength of the standing wave. b. Determine the speed of transverse waves along the string. c. The speed of waves along the string increases with increasing tension in the string. Indicate whether the value of M should be increased or decreased in order to double the number of loops in the standing wave pattern. Justify your answer. d. If a point on the string at an antinode moves a total vertical distance of 4 cm during one complete cycle, what is the amplitude of the standing wave? 7. 2004-4 (Mechanical Wave Two small speakers S are positioned a distance of 0.75 m from each other, as shown in the diagram above. The two speakers are each emitting a constant 2500 Hz tone, and the sound waves from the speakers are in phase with each other. A student is standing at point P, which is a distance of 5.0 m from the midpoint between the speakers, and hears a maximum as expected. Assume that reflections from nearby objects are negligible. Use 343 m/s for the speed of sound. a. Calculate the wavelength of these sound waves. b. The student moves a distance Y to point Q and notices that the sound intensity has decreased to a minimum. Calculate the shortest distance the student could have moved to hear this minimum. c. Identify another location on the line that passes through P and Q where the student could stand in order to observe a minimum. Justify your answer. d. i. How would your answer to (b) change if the two speakers were moved closer together? Justify your answer. ii. How would your answer to (b) change if the frequency emitted by the two speakers was increased? Justify your answer. 8. 2004b-3 (Mechanical Waves) A vibrating tuning fork is held above a column of air, as shown in the diagrams to the right. The reservoir is raised and lowered to change the water level, and thus the length of the column of air. The shortest length of air column that produces a resonance is L 1 = 0.25 m, and the next resonance is heard when the air column is L 2 = 0.80 m long. The speed of sound in air at 20 C is 343 m/s and the speed of sound in water is 1490 m/s. a. Calculate the wavelength of the standing sound wave produced by this tuning fork. b. Calculate the frequency of the tuning fork that produces the standing wave, assuming the air is at 20 C. c. Calculate the wavelength of the sound waves produced by this tuning fork in the water. d. The water level is lowered again until a third resonance is heard. Calculate the length L 3 of the air column that produces this third resonance. e. The student performing this experiment determines that the temperature of the room is actually slightly higher than 20 C. Is the calculation of the frequency in part (b) too high, too low, or still correct? Too high Too low Still correct Justify your answer.
9. 2005b-4 (Mechanical Waves) Your teacher gives you two speakers that are in phase and are emitting the same frequency of sound, which is between 5000 and 10,000 Hz. She asks you to determine this frequency more precisely. She does not have a frequency or wavelength meter in the lab, so she asks you to design an interference experiment to determine the frequency. The speed of sound is 340 m/s at the temperature of the lab room. a. From the list below, select the additional equipment you will need to do your experiment by checking the line next to each item. Speaker stand Meterstick Ruler Tape measure Stopwatch Sound-level meter b. Draw a labeled diagram of the experimental setup that you would use. On the diagram, use symbols to identify what measurements you will need to make. c. Briefly outline the procedure that you would use to make the needed measurements, including how you would use each piece of equipment you checked in a. d. Using equations, show explicitly how you would use your measurements to calculate the frequency of the sound produced by the speakers. e. If the frequency is decreased, describe how this would affect your measurements. 10. 2006-1 (Simple Harmonic Motion) 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 to the right, draw free-body diagrams showing and labeling the forces on each block when the system is in equilibrium. 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.
11. 1980-4 (Mechanical Waves, Physical Optics) In the five pairs of graphs that follow, a curve is drawn in the first graph of each pair. For the other graph in each pair, sketch the curve showing the relationship between the quantities labeled on the axes. Your graph should be consistent with the first graph in the pair.
AP Physics B Simple Harmonic Motion, Waves, Physical Optics 1977-5, two source interference (sound) c. 90 Hz 1980-4, a,b,c,d, two source and multiple source interference (light), resonance, Doppler effect 1983-3, momentum, simple harmonic motion a. 2 v 3 0 b. 2 4Mv 0 3k 1995-1, momentum, simple harmonic motion a. i. 0.60 kg m/s ii. 0.90 J b. i. 0.60 kg m/s ii. 0.40 m/s c. 0.050 m d. 0.77 s 1995-6, resonance (sound) a. 2L b. 2Lf 0 c. 2 f 0 d. L/2 1997-3, electromagnetic induction, simple harmonic motion mg mgrδd d a. b. out of page (+z direction) c. d. i 2 π 2d ε ld g ii. Δd g d 1998-5, standing waves a. 0.60 m b. 72 m/s c. M must decrease d. 1.0 cm 2004-4, 2-source interference with sound a. 0.14 m b. 0.46 m d. i. Y increases ii. Y decreases 2004b-3 resonance, standing waves with sound a. 1.1 m b. 312 Hz c. 4.8 m d. 1.35 m e. too low 2005b-4, experimental design, 2-source interference with sound e. distance between successive maxima will increase 2006-1, forces, hookean springs, free fall, simple harmonic motion b. 39 N c. 780 N/m d. 0.38 s e. 1.6 Hz f. 0.49 m/s