A comprehensive exploration of mechanical vibrations, simple harmonic motion, and the mathematical relationships between frequency, period, and amplitude.
Below are two oscillators, A (solid) and B (dashed). 1) Determine the period and frequency of each. 2) Calculate the phase difference between them in degrees. 3) State if they are in phase, out of phase, or neither.
0s 2s 4s
Problem 12: Frequency Manipulation
An industrial shaker table vibrates at a frequency of 25 Hz. A precision component must endure exactly 1,000,000 cycles for testing. How long (in hours, minutes, and seconds) must the component stay on the shaker? Show all unit conversions.
Problem 13: Phase Synthesis
Sketch two waves on the axis below: Wave A has an amplitude of 20 units and a period of 4 seconds. Wave B has an amplitude of 10 units, the same period, but is 180° out of phase with Wave A. Label the axes carefully.
Graph C
Graph D
A. Graph C has the higher frequency.
B. Graph D has the higher frequency.
C. Both graphs have the same frequency.
D. Higher frequency means a longer period.
15. Analyzing the graph below, what is the phase difference between the two vibrations?
A. 45 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
16. If a pendulum is moved from Earth to the Moon (where gravity is weaker), how does its frequency change?
A. Frequency increases
B. Frequency decreases
C. Frequency remains the same
D. The period decreases
17. Two oscillators have frequencies f and 3f. If they start in phase, how many cycles does the faster one complete before they are back in phase?
A. 1
B. 1.5
C. 3
D. 4
18. A wave has a phase of 0 rad at t=0. What is its phase at t = T/4?
A. π/4
B. π/2
C. π
D. 2π
19. Which graph represents two vibrations that are 180 degrees out of phase?
Choice A
Choice B
20. What is the phase difference between a sine wave and a cosine wave of the same frequency?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 360 degrees
21. If Oscillator X has a period of 4 seconds and Oscillator Y has a period of 2 seconds, which has the higher frequency?
A. Oscillator X
B. Oscillator Y
C. They are the same
D. Frequency is not related to period
4
Problem Set: Computational mastery
22. A pendulum completes 60 cycles in 2 minutes. Calculate the frequency (Hz) and period (s). Show your work.
23. A tuning fork vibrates at 440 Hz. How many seconds does it take for the fork to complete exactly 1,100 vibrations?
24. Sketch a vibration graph with an amplitude of 10 units and a period of 4 seconds. Draw at least two full cycles and label your axes.
25. If the length of a pendulum is decreased until its period is cut in half, what specific change happened to the frequency? Explain why.
26. Two oscillators start together. After 0.5 seconds, Oscillator A has completed 1 full cycle, while Oscillator B has completed 0.5 cycles. Determine the frequency of each and their phase difference at t = 0.5s.
27. An oscillator has a frequency of 50 Hz. How many full cycles occur in 0.5 seconds?
28. Two waves are shown below. Identify the period and amplitude of each wave and calculate the phase difference between them in degrees.
10 5 0 -5 -10 1s 2s 3s 4s 5s 6s 7s 8s Time (seconds) Amplitude Wave A Wave B
29. If two identical tuning forks are hit at different times, and Fork A is at equilibrium moving up while Fork B is at its maximum peak, what is their phase difference?
30. Energy Transformation: Describe the changes between potential energy and kinetic energy as an oscillator moves from its maximum displacement, through equilibrium, and to the opposite maximum displacement.
31. Equilibrium Points: Explain why an oscillator is at its maximum velocity when passing through the equilibrium position, using force or energy principles.
32. Define 'Equilibrium' in the context of harmonic motion and explain why it is the point of zero net force.
Graph C
Graph D
A. Graph C has the higher frequency.
B. Graph D has the higher frequency.
C. Both graphs have the same frequency.
D. Frequency is the same as the period.
15. Analyzing the graph below, what is the phase difference between the two vibrations?
A. 45 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
16. If a pendulum is moved from Earth to the Moon (where gravity is weaker), how does its frequency change?
A. Frequency increases
B. Frequency decreases
C. Frequency remains constant
D. The period decreases
17. Two oscillators have frequencies f and 3f. If they start in phase, how many cycles does the faster one complete before they are back in phase?
A. 1
B. 1.5
C. 3
D. 4
18. A wave has a phase of 0 rad at t=0. What is its phase at t = T/4?
A. π/4
B. π/2
C. π
D. 2π
19. Which graph represents two vibrations that are 180 degrees out of phase?
Choice A
Choice B
20. What is the phase difference between a sine wave and a cosine wave of the same frequency?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 360 degrees
21. If Oscillator X has a period of 4 seconds and Oscillator Y has a period of 2 seconds, which has the higher frequency?
A. Oscillator X
B. Oscillator Y
C. They are the same
D. Frequency is independent of period
22. Two vibrations reach their equilibrium points simultaneously moving in opposite directions. What is their phase difference?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
23. If an oscillator is at its maximum negative displacement at t=0, what is its phase relative to a standard sine wave?
A. 90 degrees
B. 180 degrees
C. 270 degrees
D. 360 degrees
24. Two pendulums have lengths L and 4L. What is the ratio of their periods (T_short : T_long)?
A. 1 : 1
B. 1 : 2
C. 1 : 4
D. 2 : 1
25. If two pendulums are 'in phase', they must have:
A. Identical frequencies and amplitudes
B. Identical frequencies and displacements
C. Different lengths but the same mass
D. Frequencies that are multiples of each other
26. A vibration with period T starts at equilibrium moving up. At what time t is it at its first negative peak?
A. T/4
B. T/2
C. 3T/4
D. T
27. A phase shift of π rad corresponds to what percentage of a full cycle?
A. 25%
B. 50%
C. 75%
D. 100%
4
Problem Set: Computational mastery
28. Identify the period and amplitude of each wave below. Then, calculate the phase difference in degrees. Wave B has an amplitude of 10 cm.
10 5 0 -5 -10 1s 2s 4s 6s 8s Time (seconds) Amplitude (cm) Wave A Wave B
29. A pendulum completes 75 cycles in exactly 3 minutes. Determine the frequency in Hz and the period in seconds. Show your work clearly.
30. A vibration completes 10 full cycles in 5 seconds. Which of the following is true?
A. The frequency is 0.5 Hz.
B. The period is 2.0 seconds.
C. The frequency is 2.0 Hz.
D. Both A and B are true.
C. Both graphs have the same amplitude.
D. Amplitude cannot be determined.
14. Observe Graphs C and D below. Which graph represents the vibration with the higher frequency?
Graph C
Graph D
A. Graph C has the higher frequency.
B. Graph D has the higher frequency.
C. Both graphs have the same frequency.
D. Frequency is independent of amplitude.
15. Analyzing the graph below, what is the phase difference between the two vibrations?
A. 45 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
16. If a pendulum is moved from Earth to the Moon (where gravity is weaker), how does its frequency change?
A. Frequency increases
B. Frequency decreases
C. Frequency remains constant
D. The mass must be increased
17. Two oscillators have frequencies f and 3f. If they start in phase, how many cycles does the faster one complete before they are back in phase?
A. 1
B. 1.5
C. 3
D. 4
18. A wave has a phase of 0 rad at t=0. What is its phase at t = T/4?
A. π/4
B. π/2
C. π
D. 2π
19. Which graph represents two vibrations that are 180 degrees out of phase?
Choice A
Choice B
20. What is the phase difference between a sine wave and a cosine wave of the same frequency?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 360 degrees
21. If Oscillator X has a period of 4 seconds and Oscillator Y has a period of 2 seconds, which has the higher frequency?
A. Oscillator X
B. Oscillator Y
C. They are the same
D. Frequency is independent of period
22. Two vibrations reach their equilibrium points simultaneously moving in opposite directions. What is their phase difference?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
23. If an oscillator is at its maximum negative displacement at t=0, what is its phase relative to a standard sine wave?
A. 90 degrees
B. 180 degrees
C. 270 degrees
D. 360 degrees
24. Two pendulums have lengths L and 4L. What is the ratio of their periods (T_short : T_long)?
A. 1 : 1
B. 1 : 2
C. 1 : 4
D. 2 : 1
25. If two pendulums are 'in phase', they must have:
A. Identical frequencies and amplitudes
B. Identical frequencies and displacements
C. Different lengths but the same mass
D. Frequencies that are multiples of each other
26. A vibration with period T starts at equilibrium moving up. At what time t is it at its first negative peak?
A. T/4
B. T/2
C. 3T/4
D. T
27. A phase shift of π rad corresponds to what percentage of a full cycle?
A. 25%
B. 50%
C. 75%
D. 100%
28. A vibration completes 10 full cycles in 5 seconds. Which of the following is true?
A. The frequency is 0.5 Hz.
B. The period is 2.0 seconds.
C. The frequency is 2.0 Hz.
D. Both A and B are true.
4
Problem Set: Computational mastery
29. Identify the period and amplitude of each wave below. Then, calculate the phase difference in degrees. Wave B has an amplitude of 10 cm.
100-10 2s4s Time (seconds) Amplitude (cm)
30. A pendulum completes 75 cycles in exactly 3 minutes. Determine the frequency in Hz and the period in seconds. Show your work clearly.
C. Both graphs have the same amplitude.
D. Amplitude cannot be determined.
14. Observe Graphs C and D below. Which graph represents the vibration with the higher frequency?
Graph C
Graph D
A. Graph C has the higher frequency.
B. Graph D has the higher frequency.
C. Both graphs have the same frequency.
D. Frequency is independent of amplitude.
15. Analyzing the graph below, what is the phase difference between the two vibrations?
A. 45 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
16. If a pendulum is moved from Earth to the Moon (where gravity is weaker), how does its frequency change?
A. Frequency increases
B. Frequency decreases
C. Frequency remains constant
D. The period decreases
17. Two oscillators have frequencies f and 3f. If they start in phase, how many cycles does the faster one complete before they are back in phase?
A. 1
B. 1.5
C. 3
D. 4
18. A wave has a phase of 0 rad at t=0. What is its phase at t = T/4?
A. π/4
B. π/2
C. π
D. 2π
19. Which graph represents two vibrations that are 180 degrees out of phase?
Choice A
Choice B
20. What is the phase difference between a sine wave and a cosine wave of the same frequency?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 360 degrees
21. If Oscillator X has a period of 4 seconds and Oscillator Y has a period of 2 seconds, which has the higher frequency?
A. Oscillator X
B. Oscillator Y
C. They are the same
D. Frequency is independent of period
22. Two vibrations reach their equilibrium points simultaneously moving in opposite directions. What is their phase difference?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
23. If an oscillator is at its maximum negative displacement at t=0, what is its phase relative to a standard sine wave?
A. 90 degrees
B. 180 degrees
C. 270 degrees
D. 360 degrees
24. Two pendulums have lengths L and 4L. What is the ratio of their periods (T_short : T_long)?
A. 1 : 1
B. 1 : 2
C. 1 : 4
D. 2 : 1
25. If two pendulums are 'in phase', they must have:
A. Identical frequencies and amplitudes
B. Identical frequencies and displacements
C. Different lengths but the same mass
D. Frequencies that are multiples of each other
26. A vibration with period T starts at equilibrium moving up. At what time t is it at its first negative peak?
A. T/4
B. T/2
C. 3T/4
D. T
27. A phase shift of π rad corresponds to what percentage of a full cycle?
A. 25%
B. 50%
C. 75%
D. 100%
28. A vibration completes 10 full cycles in 5 seconds. Which of the following is true?
A. The frequency is 0.5 Hz.
B. The period is 2.0 seconds.
C. The frequency is 2.0 Hz.
D. Both A and B are true.
4
Problem Set: Computational mastery
29. Identify the period and amplitude of each wave below. Then, calculate the phase difference between them in degrees. Wave B has an amplitude of 10 cm.
30. If the length of a pendulum is decreased until its period is cut in half, what specific change happened to the frequency? Explain why.
31. An oscillator has a frequency of 50 Hz. How many full cycles occur in 0.5 seconds?
32. If two identical tuning forks are hit at different times, and Fork A is at equilibrium moving up while Fork B is at its maximum peak, what is their phase difference?
33. Sketch a vibration graph with an amplitude of 10 units and a period of 4 seconds. Draw at least two full cycles and label your axes clearly.
C. Both graphs have the same amplitude.
D. Amplitude cannot be determined.
14. Observe Graphs C and D below. Which graph represents the vibration with the higher frequency?
Graph C
Graph D
A. Graph C has the higher frequency.
B. Graph D has the higher frequency.
C. Both graphs have the same frequency.
D. Frequency is independent of amplitude.
15. Analyzing the graph below, what is the phase difference between the two vibrations?
A. 45 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
16. If a pendulum is moved from Earth to the Moon (where gravity is weaker), how does its frequency change?
A. Frequency increases
B. Frequency decreases
C. Frequency remains constant
D. The period decreases
17. Two oscillators have frequencies f and 3f. If they start in phase, how many cycles does the faster one complete before they are back in phase?
A. 1
B. 1.5
C. 3
D. 4
18. A wave has a phase of 0 rad at t=0. What is its phase at t = T/4?
A. π/4
B. π/2
C. π
D. 2π
19. Which graph represents two vibrations that are 180 degrees out of phase?
Choice A
Choice B
20. What is the phase difference between a sine wave and a cosine wave of the same frequency?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 360 degrees
21. If Oscillator X has a period of 4 seconds and Oscillator Y has a period of 2 seconds, which has the higher frequency?
A. Oscillator X
B. Oscillator Y
C. They are the same
D. Frequency is independent of period
22. Two vibrations reach their equilibrium points simultaneously moving in opposite directions. What is their phase difference?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
23. If an oscillator is at its maximum negative displacement at t=0, what is its phase relative to a standard sine wave?
A. 90 degrees
B. 180 degrees
C. 270 degrees
D. 360 degrees
24. Two pendulums have lengths L and 4L. What is the ratio of their periods (T_short : T_long)?
A. 1 : 1
B. 1 : 2
C. 1 : 4
D. 2 : 1
25. If two pendulums are 'in phase', they must have:
A. Identical frequencies and amplitudes
B. Identical frequencies and displacements
C. Different lengths but the same mass
D. Frequencies that are multiples of each other
26. A vibration with period T starts at equilibrium moving up. At what time t is it at its first negative peak?
A. T/4
B. T/2
C. 3T/4
D. T
27. A phase shift of π rad corresponds to what percentage of a full cycle?
A. 25%
B. 50%
C. 75%
D. 100%
28. A vibration completes 10 full cycles in 5 seconds. Which of the following is true?
A. The frequency is 0.5 Hz.
B. The period is 2.0 seconds.
C. The frequency is 2.0 Hz.
D. Both A and B are true.
4
Problem Set: Computational mastery
29. Identify the period and amplitude of each wave below. Then, calculate the phase difference between them in degrees. Wave B has an amplitude of 10 cm.
30. If the length of a pendulum is decreased until its period is cut in half, what specific change happened to the frequency? Explain why.
31. An oscillator has a frequency of 50 Hz. How many full cycles occur in 0.5 seconds?
32. If two identical tuning forks are hit at different times, and Fork A is at equilibrium moving up while Fork B is at its maximum peak, what is their phase difference?
33. Sketch a vibration graph with an amplitude of 10 units and a period of 4 seconds. Draw at least two full cycles and label your axes clearly.
C. Both graphs have the same amplitude.
D. Amplitude cannot be determined.
14. Observe Graphs C and D below. Which graph represents the vibration with the higher frequency?
Graph C
Graph D
A. Graph C has the higher frequency.
B. Graph D has the higher frequency.
C. Both graphs have the same frequency.
D. Frequency is independent of amplitude.
15. Analyzing the graph below, what is the phase difference between the two vibrations?
A. 45 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
16. If a pendulum is moved from Earth to the Moon (where gravity is weaker), how does its frequency change?
A. Frequency increases
B. Frequency decreases
C. Frequency remains constant
D. The period decreases
17. Two oscillators have frequencies f and 3f. If they start in phase, how many cycles does the faster one complete before they are back in phase?
A. 1
B. 1.5
C. 3
D. 4
18. A wave has a phase of 0 rad at t=0. What is its phase at t = T/4?
A. π/4
B. π/2
C. π
D. 2π
19. Which graph represents two vibrations that are 180 degrees out of phase?
Choice A
Choice B
20. What is the phase difference between a sine wave and a cosine wave of the same frequency?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 360 degrees
21. If Oscillator X has a period of 4 seconds and Oscillator Y has a period of 2 seconds, which has the higher frequency?
A. Oscillator X
B. Oscillator Y
C. They are the same
D. Frequency is independent of period
22. Two vibrations reach their equilibrium points simultaneously moving in opposite directions. What is their phase difference?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
23. If an oscillator is at its maximum negative displacement at t=0, what is its phase relative to a standard sine wave?
A. 90 degrees
B. 180 degrees
C. 270 degrees
D. 360 degrees
24. Two pendulums have lengths L and 4L. What is the ratio of their periods (T_short : T_long)?
A. 1 : 1
B. 1 : 2
C. 1 : 4
D. 2 : 1
25. If two pendulums are 'in phase', they must have:
A. Identical frequencies and amplitudes
B. Identical frequencies and displacements
C. Different lengths but the same mass
D. Frequencies that are multiples of each other
26. A vibration with period T starts at equilibrium moving up. At what time t is it at its first negative peak?
A. T/4
B. T/2
C. 3T/4
D. T
27. A phase shift of π rad corresponds to what percentage of a full cycle?
A. 25%
B. 50%
C. 75%
D. 100%
28. A vibration completes 10 full cycles in 5 seconds. Which of the following is true?
A. The frequency is 0.5 Hz.
B. The period is 2.0 seconds.
C. The frequency is 2.0 Hz.
D. Both A and B are true.
4
Problem Set: Computational mastery
29. Identify the period and amplitude of each wave below. Then, calculate the phase difference between them in degrees. Wave B has an amplitude of 10 cm.
30. If the length of a pendulum is decreased until its period is cut in half, what specific change happened to the frequency? Explain why.
31. An oscillator has a frequency of 50 Hz. How many full cycles occur in 0.5 seconds?
32. If two identical tuning forks are hit at different times, and Fork A is at equilibrium moving up while Fork B is at its maximum peak, what is their phase difference?
33. Sketch a vibration graph with an amplitude of 10 units and a period of 4 seconds. Draw at least two full cycles and label your axes clearly.
C. Both graphs have the same amplitude.
D. Amplitude cannot be determined.
14. Observe Graphs C and D below. Which graph represents the vibration with the higher frequency?
Graph C
Graph B
A. Graph C has the higher frequency.
B. Graph D has the higher frequency.
C. Both graphs have the same frequency.
D. Frequency is independent of amplitude.
15. Analyzing the graph below, what is the phase difference between the two vibrations?
A. 45 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
16. If a pendulum is moved from Earth to the Moon (where gravity is weaker), how does its frequency change?
A. Frequency increases
B. Frequency decreases
C. Frequency remains constant
D. The period decreases
17. Two oscillators have frequencies f and 3f. If they start in phase, how many cycles does the faster one complete before they are back in phase?
A. 1
B. 1.5
C. 3
D. 4
18. A wave has a phase of 0 rad at t=0. What is its phase at t = T/4?
A. π/4
B. π/2
C. π
D. 2π
19. Which graph represents two vibrations that are 180 degrees out of phase?
Choice A
Choice B
20. What is the phase difference between a sine wave and a cosine wave of the same frequency?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 360 degrees
21. If Oscillator X has a period of 4 seconds and Oscillator Y has a period of 2 seconds, which has the higher frequency?
A. Oscillator X
B. Oscillator Y
C. They are the same
D. Frequency is independent of period
22. Two vibrations reach their equilibrium points simultaneously moving in opposite directions. What is their phase difference?
A. 0 degrees
B. 90 degrees
C. 180 degrees
D. 270 degrees
23. If an oscillator is at its maximum negative displacement at t=0, what is its phase relative to a standard sine wave?
A. 90 degrees
B. 180 degrees
C. 270 degrees
D. 360 degrees
24. Two pendulums have lengths L and 4L. What is the ratio of their periods (T_short : T_long)?
A. 1 : 1
B. 1 : 2
C. 1 : 4
D. 2 : 1
25. If two pendulums are 'in phase', they must have:
A. Identical frequencies and amplitudes
B. Identical frequencies and displacements
C. Different lengths but the same mass
D. Frequencies that are multiples of each other
26. A vibration with period T starts at equilibrium moving up. At what time t is it at its first negative peak?
A. T/4
B. T/2
C. 3T/4
D. T
27. A phase shift of π rad corresponds to what percentage of a full cycle?
A. 25%
B. 50%
C. 75%
D. 100%
28. A vibration completes 10 full cycles in 5 seconds. Which of the following is true?
A. The frequency is 0.5 Hz.
B. The period is 2.0 seconds.
C. The frequency is 2.0 Hz.
D. Both A and B are true.
4
Problem Set: Computational mastery
29. Identify the period and amplitude of each wave below. Then, calculate the phase difference between them in degrees. Wave B has an amplitude of 10 cm.
30. If the length of a pendulum is decreased until its period is cut in half, what specific change happened to the frequency? Explain why.
31. An oscillator has a frequency of 50 Hz. How many full cycles occur in 0.5 seconds?
32. If two identical tuning forks are hit at different times, and Fork A is at equilibrium moving up while Fork B is at its maximum peak, what is their phase difference?
33. Sketch a vibration graph with an amplitude of 10 units and a period of 4 seconds. Draw at least two full cycles and label your axes clearly.
