Question 1

A mass $$m$$ attached to a spring with spring constant $$k$$ oscillates with simple harmonic motion. If the amplitude of oscillation is doubled, what happens to the total mechanical energy $$E of $$the system?

Accepted Answer

$$E is $$quadrupled

Question 2

A pendulum of length $$L is $$swinging with small amplitude. Which formula gives its period $$T$$?

Accepted Answer

$$T = \pi \sqrt{\dfrac{L}{g}}$$

Question 3

A string of length $$L is $$fixed at both ends. What is the wavelength $$\lambda_n of $$the $$nth $$harmonic?

Accepted Answer

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

Question 4

A sound wave travels through air at $$20^\circC. $$What is its speed?

Accepted Answer

$$v = 331 + 0.6 	imes 20$$

Question 5

A wave on a string has a speed $$v$$ given by $$v = \sqrt{\dfrac{F}{\mu}}. If $$the tension $$F is $$quadrupled, what happens to the wave speed?

Accepted Answer

It doubles

Question 6

A mass-spring system oscillates with period $$T = 2\pi \sqrt{\dfrac{m}{k}}. If $$the mass is increased by a factor of 4, what happens to the period?

Accepted Answer

It doubles

Question 7

A standing wave on a string is described by $$y(x, t) = 2A \sin(kx) \cos(\omega t). $$What is the distance between adjacent nodes?

Accepted Answer

$$\dfrac{\lambda}{2}$$

Question 8

A pipe is open at both ends. What is the fundamental frequency $$f_1 if $$the speed of sound is $$v$$ and the length of the pipe is $$L$$?

Accepted Answer

$$f_1$$ = \dfrac{v}{2L}$$

Question 9

A wave has an intensity $$I$$ and passes through an area $$A. $$What is the formula for intensity?

Accepted Answer

$$I = \dfrac{P}{A}$$

Question 10

A sound source emits a power $$P$$ uniformly in all directions. At a distance $$r$$ from the source, what is the intensity $$I$$?

Accepted Answer

$$I = \dfrac{P}{4\pi r^2$}$$

Question 11

A wave on a string has amplitude $$A$$, angular frequency $$\omega$$, and mass density $$\mu. $$What is the power transmitted by the wave?

Accepted Answer

$$p = \dfrac{1}{2} \mu \$$omega^2$$ A^2$

Question 12

A damped oscillator has a displacement $$x(t) = A_0$ e$$^{-bt/2m}$$ \cos(\omega' t). $$What does the term $$e^{-bt/2m}$$ represent?

Accepted Answer

The exponential decay of amplitude due to damping

Question 13

A closed pipe (one end closed) of length $$L$$ supports only odd harmonics. What is the frequency of the third harmonic?

Accepted Answer

$$f_3$$ = \dfrac{3v}{4L}$$

Question 14

A traveling wave is described by $$y(x, t) = A \sin(kx - \omega t). $$What is the direction of propagation?

Accepted Answer

To the right

Question 15

The Doppler effect formula for a moving observer and source is $$f' = f \dfrac{v \pm v_o}{v \mp v_s}. If $$the observer moves toward a stationary source, which sign should be used in the numerator?

Accepted Answer

Plus

Question 16

A string has a linear mass density $$\mu$$ and is under tension $$F. If$$ $$\mu is $$doubled, what happens to the speed of a wave on the string?

Accepted Answer

It decreases by a factor of $$\sqrt{2}$$

Question 17

The intensity level in decibels is given by $$\beta = 10 \log \left( \dfrac{I}{I_0$} \right). If $$the intensity increases by a factor of 100, how much does $$\beta$$ increase?

Accepted Answer

By 20 dB

Question 18

A mass-spring system has angular frequency $$\omega = \sqrt{\dfrac{k}{m}}. If $$the spring constant $$k is $$increased by a factor of 9, what happens to $$\omega$$?

Accepted Answer

It triples

Question 19

A standing wave on a string has antinode separation $$\dfrac{\lambda}{2}. If $$the string is 1.5 m long and supports a standing wave with 3 antinodes, what is the wavelength?

Accepted Answer

1 m

Vibrations & Waves: Key Formulas and Concepts

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