Question 1

Which of the following best describes the equation for the displacement $$x(t) of a $$simple harmonic oscillator, as shown in the first image?

Accepted Answer

$$x(t) = A \cos(\omega t + \phi)$$

Question 2

In the second image, a mass $$m$$ attached to a spring is shown in three positions. What is the direction of the restoring force $$\vec{F}_s$$ when the mass is displaced to the right of equilibrium ($$x \u003e 0$$)?

Accepted Answer

To the left, opposite to the displacement

Question 3

The equation of motion for a mass-spring system is $$m \frac{d^2$x}{dt^2$} = -kx. $$What is the angular frequency $$\omega of $$the oscillation?

Accepted Answer

$$\omega = \sqrt{\frac{k}{m}}$$

Question 4

Refer to the fourth image, which shows both displacement $$x(t)$$ and velocity $$v(t)$$ for a harmonic oscillator. At what point does the velocity reach its maximum value?

Accepted Answer

When $$x(t) is $$zero

Question 5

In the fifth image, the total energy $$E of a $$harmonic oscillator remains constant while kinetic $$K$$ and potential $$U$$ energies oscillate. Which equation correctly represents the total energy?

Accepted Answer

$$E = \frac{1}{2}kA^2$

Question 6

A wave is propagating in the positive $$x-$$direction as shown in the sixth image. Which equation best describes its wave function?

Accepted Answer

$$y(x, t) = A \sin(kx - \omega t)$$

Question 7

The seventh image shows a longitudinal wave. What distinguishes a longitudinal wave from a transverse wave?

Accepted Answer

Oscillations are parallel to the direction of propagation

Question 8

In the context of mechanical waves, what is the relationship between wave speed $$v$$, tension $$T_S$$, and mass per unit length $$\mu$$ for a string?

Accepted Answer

$$v = \sqrt{\frac{T_S}{\mu}}$$

Question 9

A standing wave is formed by the superposition of two waves traveling in opposite directions. Which equation describes the displacement $$y(x, t) of a $$standing wave?

Accepted Answer

$$y(x, t) = 2A \sin(kx) \cos(\omega t)$$

Question 10

For a string of length $$L$$ with both ends fixed, what are the allowed wavelengths $$\lambda_n$$ for its normal modes?

Accepted Answer

$$\lambda_n = \frac{2L}{n}$$

Question 11

A sound wave in air has a speed $$v$$ given by $$v = \sqrt{\gamma k_B T / m}. $$What does $$\gamma$$ represent in this equation?

Accepted Answer

The adiabatic index (heat capacity ratio)

Question 12

In a tube with one end open and one end closed, what is the fundamental frequency $$f_1 of $$the standing sound wave?

Accepted Answer

$$f_1$$ = \frac{v}{4L}$$

Question 13

The intensity $$I of a $$sound wave is related to its pressure amplitude $$\Delta P_{max}$$ and the speed of sound $$v by $$which equation?

Accepted Answer

$$I = \frac{v}{2B} \Delta P_{max}^2$$

Question 14

If two sound waves of frequencies $$f_1$$ and $$f_2$$ interfere, what is the beat frequency $$f_b$$ observed?

Accepted Answer

$$f_b = |f_1$ - f_2$|$$

Question 15

A piano tuner hears 2 beats per second when tuning a string to 523 Hz. What are the possible frequencies of the string?

Accepted Answer

521 Hz or 525 Hz

Question 16

In a standing sound wave in a tube, what is the relationship between displacement nodes and pressure antinodes?

Accepted Answer

Displacement nodes coincide with pressure antinodes

Question 17

A string of length $$L$$ and mass $$m is $$under tension $$T. $$What is the fundamental frequency $$f_1 of $$its vibration?

Accepted Answer

$$f_1$$ = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$$

Question 18

Which of the following best describes constructive interference between two waves?

Accepted Answer

The waves are in phase by $$2\pi n$$

Question 19

A wave pulse travels along a string with mass per unit length $$\mu$$ and tension $$T_S. If $$the string is 20 m long and the tension increases with height, how does the wave speed $$v(z)$$ depend on the distance $$z$$ from the lower end?

Accepted Answer

$$v(z) = \sqrt{\frac{T_S(z)}{\mu}}$$

Oscillatory Motion and Mechanical Waves: Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards