Q: If the frequency of the second harmonic is $$900\mathrm{\ Hz}$$ for a string of length $$2\mathrm{\ m}$$, what is the wave speed $$v on $$the string?

$$v = \lambda f = 2\mathrm{\ m} \times 900\mathrm{\ Hz} = 1800\mathrm{\ m/s}$$

Question 1

A string of length $$L = 2\mathrm{\ m} is $$under tension $$F_T = 16.2\mathrm{\ N}$$ and has a second harmonic frequency of $$f_2 = 900$$\mathrm{\ Hz}$$. What is the mass per unit length $$\mu$$ of the string?

Accepted Answer

$$\mu = \dfrac{F_T}{4L^2$f_2^2$}$$

Question 2

Given a string fixed at both ends, which of the following correctly describes the wavelength $$\lambda of $$the second harmonic?

Accepted Answer

$$\lambda = L$$

Question 3

If the tension in a string is doubled, how does the fundamental frequency $$f_1$$ change? Assume all other parameters remain constant.

Accepted Answer

$$f_1$$ increases by a factor of $$\sqrt{2}$$

Question 4

A student plucks a string and observes a standing wave with three antinodes. What is the harmonic number $$n$$?

Accepted Answer

$$n = 3$$

Question 5

Which equation gives the frequency of the $$nth $$harmonic for a string of length $$L$$, tension $$F_T$$, and mass per unit length $$\mu$$?

Accepted Answer

$$f_n = \dfrac{n}{2L} \sqrt{\dfrac{F_T}{\mu}}$$

Question 6

A guitar string of length $$0.65\mathrm{\ m} is $$under tension $$120\mathrm{\ N}$$ and has a mass per unit length $$0.005\mathrm{\ kg/m}. $$What is the speed of waves on the string?

Accepted Answer

$$v = \sqrt{\dfrac{120}{0.005}} = 155.0\mathrm{\ m/s}$$

Question 7

If the frequency of the second harmonic is $$900\mathrm{\ Hz}$$ for a string of length $$2\mathrm{\ m}$$, what is the wave speed $$v on $$the string?

Accepted Answer

$$v = \lambda f = 2\mathrm{\ m} 	imes 900\mathrm{\ Hz} = 1800\mathrm{\ m/s}$$

Question 8

A string vibrates in its second harmonic. Which of the following best describes the number of nodes and antinodes?

Accepted Answer

3 nodes, 2 antinodes

Question 9

A violin string is tuned to a higher pitch by increasing the tension. What effect does this have on the wave speed and frequency?

Accepted Answer

Both wave speed and frequency increase

Question 10

A string of length $$L$$ and mass $$m is $$under tension $$F_T. $$What is the correct expression for the mass per unit length $$\mu$$?

Accepted Answer

$$\mu = \dfrac{m}{L}$$

Question 11

A student wants to double the fundamental frequency of a string. Which of the following changes will achieve this?

Accepted Answer

Decrease the length $$L by $$half

Question 12

Which of the following is a correct statement about standing waves on a string fixed at both ends?

Accepted Answer

Nodes occur at the fixed ends

Question 13

A string of length $$L is $$vibrating in its third harmonic. What is the wavelength $$\lambda of $$the standing wave?

Accepted Answer

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

Question 14

If the mass of a string is increased while keeping the length and tension constant, what happens to the fundamental frequency?

Accepted Answer

It decreases

Question 15

A string of length $$L is $$vibrating in its fundamental mode. What is the distance between two consecutive nodes?

Accepted Answer

$$L$$

Question 16

A piano wire is $$1.5\mathrm{\ m}$$ long, has a mass of $$0.03\mathrm{\ kg}$$, and is under a tension of $$400\mathrm{\ N}. $$What is the fundamental frequency?

Accepted Answer

$$f_1$$ = \dfrac{1}{2L} \sqrt{\dfrac{F_T}{\mu}} = \dfrac{1}{3} \sqrt{\dfrac{400}{0.02}} \approx 47.1\mathrm{\ Hz}$$

Standing Waves on a String: Harmonics, Frequency, and Mass Calculation

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