Question 1

A string of mass 5 g and length 1.25 m is tightly attached to a mechanical vibrator at one end and to a fixed support at the other end. The mechanical vibrator generates traveling waves modeled by the wave function $$y = 4 	ext{ mm} \sin(2\pi x + 80\pi t). $$What is the tension $$T in $$the string and the average power $$P_{av}$$ transmitted by the traveling wave?

Accepted Answer

$$i)\ T = 8\ N;\ ii)\ P_{av} = 0.101\ W$$

Question 2

Waves travel at $$225\ 	ext{m/s} on a $$strained wire. If the tension is adjusted to $$\frac{1}{5} of $$the initial value, what is the new speed of the waves on the wire?

Accepted Answer

$$6.71\ 	ext{m/s}$$

Question 3

A guitar string of length $$0.65\ 	ext{m}$$ and mass $$3.9\ 	ext{g} is $$under a tension of $$120\ 	ext{N}. $$What is the speed of a wave traveling along the string?

Accepted Answer

$$279\ 	ext{m/s}$$

Question 4

A wave on a string is described by $$y(x, t) = 0.02\ 	ext{m} \sin(4\pi x - 200\pi t)$$, where $$x$$ and $$y$$ are in meters and $$t in $$seconds. What is the frequency of the wave?

Accepted Answer

$$100\ 	ext{Hz}$$

Question 5

A $$2.5\ 	ext{m}$$ long wire has a mass of $$0.015\ 	ext{kg}$$ and is under a tension of $$60\ 	ext{N}. $$What is the linear mass density $$\mu of $$the wire?

Accepted Answer

$$0.006\ 	ext{kg/m}$$

Question 6

A wave pulse travels down a string with speed $$v. If $$the tension in the string is quadrupled, what happens to the speed of the wave?

Accepted Answer

It doubles: $$v_{new} = 2v$$

Question 7

A $$0.5\ 	ext{kg}$$ mass is attached to a string and oscillates with a frequency of $$2\ 	ext{Hz}. $$What is the period of oscillation?

Accepted Answer

$$0.5\ 	ext{s}$$

Question 8

A wave on a string has amplitude $$A$$, angular frequency $$\omega$$, and linear mass density $$\mu. $$The average power transmitted by the wave is given by:

Accepted Answer

$$P_{av} = \frac{1}{2} \mu \$$omega^2$$ A^2$ v$$

Question 9

A wave travels along a string with speed $$v = \sqrt{\frac{T}{\mu}}. If $$the tension $$T is$$ $$16\ 	ext{N}$$ and the linear mass density $$\mu is$$ $$0.004\ 	ext{kg/m}$$, what is the speed of the wave?

Accepted Answer

$$63.25\ 	ext{m/s}$$

Question 10

A mechanical vibrator produces a wave on a string with frequency $$f = 40\ 	ext{Hz}$$ and wavelength $$\lambda = 0.5\ 	ext{m}. $$What is the speed of the wave?

Accepted Answer

$$20\ 	ext{m/s}$$

Question 11

A wave on a string is described by $$y(x, t) = 0.01\ 	ext{m} \sin(6\pi x - 120\pi t). $$What is the wavelength of the wave?

Accepted Answer

$$\frac{1}{3}\ 	ext{m}$$

Question 12

A $$10\ 	ext{m}$$ long wire has a mass of $$0.02\ 	ext{kg}$$ and is under a tension of $$50\ 	ext{N}. $$What is the speed of a wave on the wire?

Accepted Answer

$$50\ 	ext{m/s}$$

Question 13

A wave pulse on a string has amplitude $$A = 0.005\ 	ext{m}$$, angular frequency $$\omega = 100\ 	ext{rad/s}$$, and speed $$v = 20\ 	ext{m/s}. $$The string has linear mass density $$\mu = 0.002\ 	ext{kg/m}. $$What is the average power transmitted by the wave?

Accepted Answer

$$0.05\ 	ext{W}$$

Question 14

A wave on a string is described by $$y(x, t) = 0.03\ 	ext{m} \sin(2\pi x - 60\pi t). $$What is the period of the wave?

Accepted Answer

$$0.033\ 	ext{s}$$

Question 15

A $$1.5\ 	ext{m}$$ long string has a mass of $$0.012\ 	ext{kg}$$ and is under a tension of $$18\ 	ext{N}. $$What is the speed of a wave on the string?

Accepted Answer

$$50\ 	ext{m/s}$$

Question 16

A wave on a string has a frequency of $$60\ 	ext{Hz}$$ and a wavelength of $$0.4\ 	ext{m}. $$What is the speed of the wave?

Accepted Answer

$$24\ 	ext{m/s}$$

Question 17

A wave on a string is described by $$y(x, t) = 0.02\ 	ext{m} \sin(5\pi x - 100\pi t). $$What is the wave number $$k$$?

Accepted Answer

$$5\pi\ 	ext{rad/m}$$

Question 18

A $$2\ 	ext{m}$$ long wire has a mass of $$0.01\ 	ext{kg}$$ and is under a tension of $$40\ 	ext{N}. $$What is the linear mass density $$\mu$$?

Accepted Answer

$$0.005\ 	ext{kg/m}$$

Question 19

A wave on a string is described by $$y(x, t) = 0.01\ 	ext{m} \sin(8\pi x - 160\pi t). $$What is the frequency of the wave?

Accepted Answer

$$80\ 	ext{Hz}$$

Study Notes: Mechanical Waves on Strings and Wires

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