Q: Two tuning forks produce frequencies of $$256\ \mathrm{Hz}$$ and $$260\ \mathrm{Hz}. $$What is the beat frequency heard?

$$f_{\text{beat}} = 4\ \mathrm{Hz}$$

Q: A mass on a spring oscillates with $$x(t) = 0.10 \cos(5t)$$, where $$x is in $$meters and $$t in $$seconds. What is the maximum velocity of the mass?

$$v_{\text{max}} = 0.50\ \mathrm{m/s}$$

Q: A light wave in water ($$n = 1.33) $$has a wavelength of $$450\ \mathrm{nm}. $$What is its frequency?

$$f = 5.0 \times 10$$^{14}$$\ \mathrm{Hz}$$

Question 1

A mass attached to a spring oscillates with amplitude $$A$$ and angular frequency $$\omega. At $$what displacement $$x$$ from equilibrium is the kinetic energy equal to the potential energy?

Accepted Answer

$$x = \frac{A}{\sqrt{2}}$$

Question 2

A sound wave travels through air at $$20^\circC. $$What is the speed of sound in air at this temperature?

Accepted Answer

$$v = 343\ \mathrm{m/s}$$

Question 3

A 0.5 kg mass is attached to a spring with spring constant $$k = 200\ \mathrm{N/m}. $$What is the period of oscillation?

Accepted Answer

$$T = 0.314\ \mathrm{s}$$

Question 4

A train approaches a stationary observer at $$30\ \mathrm{m/s}$$, emitting a whistle at $$500\ \mathrm{Hz}. $$What frequency does the observer hear? (Assume $$v = 340\ \mathrm{m/s})$$

Accepted Answer

$$f' = 545\ \mathrm{Hz}$$

Question 5

Which equation correctly expresses the total mechanical energy of a mass-spring system in simple harmonic motion?

Accepted Answer

$$E = \frac{1}{2}mv^2$ + \frac{1}{2}kx^2$

Question 6

A string fixed at both ends vibrates in its third harmonic. If the string length is $$L$$, what is the wavelength $$\lambda of $$the standing wave?

Accepted Answer

$$\lambda = \frac{2L}{3}$$

Question 7

A light ray passes from air ($$n_1 = 1.00$) into glass (n_2$ = 1.50) at an $$angle of $$30^\circ to $$the normal. What is the angle of refraction in the glass?

Accepted Answer

$$\$$theta_2$$ = 19.5^\circ$$

Question 8

Two tuning forks produce frequencies of $$256\ \mathrm{Hz}$$ and $$260\ \mathrm{Hz}. $$What is the beat frequency heard?

Accepted Answer

$$f_{	ext{beat}} = 4\ \mathrm{Hz}$$

Question 9

A lens forms a real image of an object placed $$30\ \mathrm{cm}$$ from the lens. The image is formed $$60\ \mathrm{cm} on $$the other side. What is the focal length of the lens?

Accepted Answer

$$f = 20\ \mathrm{cm}$$

Question 10

A mass on a spring oscillates with $$x(t) = 0.10 \cos(5t)$$, where $$x is in $$meters and $$t in $$seconds. What is the maximum velocity of the mass?

Accepted Answer

$$v_{	ext{max}} = 0.50\ \mathrm{m/s}$$

Question 11

A pipe open at both ends has a fundamental frequency of $$440\ \mathrm{Hz}. If $$the speed of sound is $$330\ \mathrm{m/s}$$, what is the length of the pipe?

Accepted Answer

$$L = 0.375\ \mathrm{m}$$

Question 12

A light wave in water ($$n = 1.33) $$has a wavelength of $$450\ \mathrm{nm}. $$What is its frequency?

Accepted Answer

$$f = 5.0 	imes 10$$^{14}$$\ \mathrm{Hz}$$

Question 13

A 2.0 kg mass oscillates on a spring with amplitude $$0.20\ \mathrm{m}$$ and $$k = 50\ \mathrm{N/m}. $$What is the total mechanical energy of the system?

Accepted Answer

$$E = 1.0\ \mathrm{J}$$

Question 14

A car horn emits a frequency of $$400\ \mathrm{Hz}. If $$the car moves away from a stationary observer at $$20\ \mathrm{m/s}$$ and the speed of sound is $$340\ \mathrm{m/s}$$, what frequency does the observer hear?

Accepted Answer

$$f' = 377\ \mathrm{Hz}$$

Question 15

A lens has a focal length of $$10\ \mathrm{cm}. An $$object is placed $$15\ \mathrm{cm}$$ from the lens. Where is the image formed?

Accepted Answer

$$s' = 30\ \mathrm{cm}$$\ (on\ the\ opposite\ side\ of\ the\ lens)$

Question 16

A string of length $$1.2\ \mathrm{m} is $$fixed at both ends. What is the frequency of the second harmonic if the wave speed is $$180\ \mathrm{m/s}$$?

Accepted Answer

$$f_2 = 150$$\ \mathrm{Hz}$$

Question 17

A light ray passes from water ($$n_1 = 1.33$) into air (n_2$ = 1.00) at an $$angle of incidence of $$60^\circ. $$What is the angle of refraction?

Accepted Answer

$$\$$theta_2$$ = 83.5^\circ$$

Question 18

A pipe closed at one end has a length of $$0.85\ \mathrm{m}. $$What is the fundamental frequency if the speed of sound is $$340\ \mathrm{m/s}$$?

Accepted Answer

$$f_1 = 100$$\ \mathrm{Hz}$$

Question 19

A mass-spring system oscillates with period $$T = 0.5\ \mathrm{s}. $$What is the angular frequency $$\omega$$?

Accepted Answer

$$\omega = 4\pi\ \mathrm{rad/s}$$

Question 20

A converging lens forms an image with a magnification of $$-2.0. If $$the object is $$15\ \mathrm{cm}$$ from the lens, where is the image?

Accepted Answer

$$s' = -30\ \mathrm{cm}$$\ (on\ the\ same\ side\ as\ the\ object)$

Oscillations, Waves, and Optics: Essential Formulas and Concepts

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