7. In the figure below, a string, tied to a sinusoidal oscillator at P and runni

Question

7. In the figure below, a string, tied to a sinusoidal oscillator at P and running over a support
at Q, is stretched by a block of mass m. Separation L = 1.20 m, linear mass density ? =
1.6 g/m, and the oscillator frequency f = 120 Hz. The amplitude of the motion at P is small
enough for that point to be considered a node. A node also exists at Q.

(a) What mass m allows the oscillator to set up the fourth harmonic on the string?
(b) What standing wave mode, if any, can be set up if m = 1.00 kg?

Please explain fully. I want to understand the problem, not just get the answer.

Explanation / Answer

For part A)

Apply (v = sqrt{T/mu})

v = f(wavelength)

u = 1.6 g/m which is 1.6 X 10^-3 kg/m

For the fourth harmonic, the wevelength is half the string length = .6 m

(120)(.6) = (T/1.6 X 10^-3)^.5

T = 8.29 N

Mass = T/g = 8.29/9.8

mas = .846 kg

Part B

v = (9.8/1.6 X 10^-3)^.5

v = 78.26 m/s

v = f(wavelength)

78.26 = (120)(wavelength)

wavelength = .652 m

Can we form a perfect wavelength of .652 from the 1.2 m long string?

Look at harmonic series

f1 has wavenength of 2.4m

f2 will be 1.2 m

f3 will be .8 m

f4 will be .6 m

If we continue, we will be getting smaller, so the .652 fits somewhere between f3 and f4, therefore it will not give a perfect standing wave.

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