Q: A 0.3 kg mass stretches a spring by 0.15 m. What is the spring constant $$k of $$the spring?

$$k = \frac{mg}{\Delta x} = \frac{0.3 \times 9.81}{0.15} = 19.6\ \mathrm{N/m}$$

Q: What is the maximum velocity $$v_{max} of $$the mass in the above spring system?

$$v_{max} = \omega A = 8.08 \times 0.1 = 0.808\ \mathrm{m/s}$$

Q: A speaker at a rock concert produces $$130\ \mathrm{dB} at$$ $$2.2\ \mathrm{m} in $$front of the speaker. What is the power output of the speaker?

$$P = 4\pi r^2$ I = 4\pi (2.2)^2$ \times 10.0 = 608\ \mathrm{W}$$

Q: A resonance tube produces a loud sound at $$L_1 = 17$$\ \mathrm{cm}$$ and L_2$ = 51\ \mathrm{cm}$$ with a $$500\ \mathrm{Hz}$$ tuning fork. What is the speed of sound in the tube?

$$v = \lambda f = 0.68 \times 500 = 340\ \mathrm{m/s}$$

Q: A supersonic jet travels at Mach 2.2 at an altitude of $$9500\ \mathrm{m}. $$What is the distance $$D$$ from the observer to the jet when the sonic boom is heard?

$$D = \frac{9500}{\tan \theta}$$, where $$\theta = \sin$$^{-1}$$\left(\frac{v_{sound}}{v_{jet}}\right) = \sin$$^{-1}$$\left(\frac{1}{2.2}\right) = 27^\circ$$, so $$D = \frac{9500}{\tan 27^\circ} = 18616\ \mathrm{m}$$

Q: A vertical spring has $$k = 115\ \mathrm{N/m}$$ and a $$75\ \mathrm{g}$$ mass. If the amplitude drops from $$5.0\ \mathrm{cm} to$$ $$2.0\ \mathrm{cm} in$$ $$3.5\ \mathrm{s}$$, what is the damping constant $$b$$?

$$b = \frac{2m}{t} \ln \frac{A_0$}{A} = \frac{2 \times 0.075}{3.5} \ln \frac{5}{2} = 0.039\ \mathrm{kg/s}$$

Question 1

A 0.3 kg mass stretches a spring by 0.15 m. What is the spring constant $$k of $$the spring?

Accepted Answer

$$k = \frac{mg}{\Delta x} = \frac{0.3 	imes 9.81}{0.15} = 19.6\ \mathrm{N/m}$$

Question 2

A mass-spring system with $$k = 19.6\ \mathrm{N/m}$$ and $$m = 0.3\ \mathrm{kg} is $$compressed by $$0.1\ \mathrm{m}$$ and released. What is the angular frequency $$\omega$$?

Accepted Answer

$$\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{19.6}{0.3}} = 8.08\ \mathrm{rad/s}$$

Question 3

What is the maximum velocity $$v_{max} of $$the mass in the above spring system?

Accepted Answer

$$v_{max} = \omega A = 8.08 	imes 0.1 = 0.808\ \mathrm{m/s}$$

Question 4

A pendulum of length $$8\ \mathrm{m} is $$released from a $$15^\circ$$ angle. What is its period $$T$$?

Accepted Answer

$$T = 2\pi \sqrt{\frac{l}{g}} = 2\pi \sqrt{\frac{8}{9.81}} = 5.67\ \mathrm{s}$$

Question 5

A wire of length $$8\ \mathrm{m}$$ and mass $$3.6\ \mathrm{kg} is $$under tension $$14\ \mathrm{N}. $$What is the speed $$v of a $$wave pulse on the wire?

Accepted Answer

$$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{14}{0.45}} = 5.58\ \mathrm{m/s}$$

Question 6

An observer moves at $$20\ \mathrm{m/s}$$ toward a stationary siren emitting $$440\ \mathrm{Hz}. $$What frequency does the observer hear? (Speed of sound $$v = 343\ \mathrm{m/s})$$

Accepted Answer

$$f = f_s \frac{v + v_o}{v} = 440 \frac{343 + 20}{343} = 466\ \mathrm{Hz}$$

Question 7

A speaker at a rock concert produces $$130\ \mathrm{dB} at$$ $$2.2\ \mathrm{m} in $$front of the speaker. What is the power output of the speaker?

Accepted Answer

$$P = 4\pi r^2$ I = 4\pi (2.2)^2$ 	imes 10.0 = 608\ \mathrm{W}$$

Question 8

A resonance tube produces a loud sound at $$L_1 = 17$$\ \mathrm{cm}$$ and L_2$ = 51\ \mathrm{cm}$$ with a $$500\ \mathrm{Hz}$$ tuning fork. What is the speed of sound in the tube?

Accepted Answer

$$v = \lambda f = 0.68 	imes 500 = 340\ \mathrm{m/s}$$

Question 9

A supersonic jet travels at Mach 2.2 at an altitude of $$9500\ \mathrm{m}. $$What is the distance $$D$$ from the observer to the jet when the sonic boom is heard?

Accepted Answer

$$D = \frac{9500}{	an 	heta}$$, where $$	heta = \sin$$^{-1}$$\left(\frac{v_{sound}}{v_{jet}}ight) = \sin$$^{-1}$$\left(\frac{1}{2.2}ight) = 27^\circ$$, so $$D = \frac{9500}{	an 27^\circ} = 18616\ \mathrm{m}$$

Question 10

A vertical spring has $$k = 115\ \mathrm{N/m}$$ and a $$75\ \mathrm{g}$$ mass. If the amplitude drops from $$5.0\ \mathrm{cm} to$$ $$2.0\ \mathrm{cm} in$$ $$3.5\ \mathrm{s}$$, what is the damping constant $$b$$?

Accepted Answer

$$b = \frac{2m}{t} \ln \frac{A_0$}{A} = \frac{2 	imes 0.075}{3.5} \ln \frac{5}{2} = 0.039\ \mathrm{kg/s}$$

Question 11

A ball of mass $$8\ \mathrm{kg} is $$attached to a $$3\ \mathrm{m}$$ string and released from $$15^\circ. $$What is its velocity at the lowest point?

Accepted Answer

$$v = \sqrt{2gh}$$, where $$h = L(1 - \cos 	heta) = 3(1 - \cos 15^\circ) = 0.102\ \mathrm{m}$$, so $$v = \sqrt{2 	imes 9.81 	imes 0.102} = 1.42\ \mathrm{m/s}$$

Question 12

A child drops pebbles into a pond, creating ripples with a wavelength of $$6.0\ \mathrm{m}$$ and 30 crests per minute. What is the frequency of the ripples?

Accepted Answer

$$f = \frac{30}{60} = 0.5\ \mathrm{Hz}$$

Question 13

A wave on a wire has a wavelength $$0.30\ \mathrm{m}$$, tension $$1000\ \mathrm{N}$$, total mass $$15\ \mathrm{kg}$$, and length $$300\ \mathrm{m}. $$What is the frequency?

Accepted Answer

$$f = \frac{v}{\lambda}$$, $$v = \sqrt{\frac{F_T}{\mu}} = \sqrt{\frac{1000}{15/300}} = 141\ \mathrm{m/s}$$, so $$f = \frac{141}{0.3} = 470\ \mathrm{Hz}$$

Question 14

Two sound waves in a gas have wavelengths $$2.0\ \mathrm{m}$$ and $$2.1\ \mathrm{m}$$, producing 8 beats per second. What is the velocity of sound?

Accepted Answer

$$v = 8 	imes \frac{2.1 	imes 2.0}{2.1 - 2.0} = 336\ \mathrm{m/s}$$

Question 15

A standing wave on a string with both ends fixed has length $$L$$ and wavelength $$\lambda. $$What is the relationship for the fundamental mode?

Accepted Answer

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

Question 16

An open organ pipe emits middle C ($$262\ \mathrm{Hz}) at$$ $$15^\circC. $$What is the length of the pipe?

Accepted Answer

$$l = \frac{v}{2f} = \frac{340}{2 	imes 262} = 0.649\ \mathrm{m}$$

PHYS225 - Exam #1 Study Guide (Oscillations, Waves, and Sound)

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