In the figure, a string, tied to a sinusoidal oscillator at P and running over a

Question

In the figure, 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 = 1.5 m, linear density mu = 1.4 g/m, and the oscillator frequency f = 100 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 = 4 kg (Give 0 if the mass cannot set up a standing wave)? (a) Number Units (b) Number Units

Explanation / Answer

Given, lengtho f the string be L = 1.5 m

linar density of the string, mu = 1.4 g/m = 0.0014 kg/m

osscilator frequency, f = 100 HZ

a) for the fourth harmonic

2*lambda = L

lambda = L/2

but speed of wave, v = lambda*f

but v = sqroot(T/mu)

where T is tension in the string

so, sqroot(T/mu) = lambda*f = Lf/2

T = mu*L^2*f^2/4

but T = mg ( from force balance)

so, m = mu*L^2*f^2/4g = 0.0014*1.5^2*100^2/4*9.81 = 0.8027 kg

b) at m = 4 kg

tension in the string = 4g

wave speed on string, v = swroot(T/mu) = sqroot(4g/mu) = 167.4173 m/s

now f = 100 Hz

so for any wavelength

lambda*f = v

lambda = 1.674 m

also, L = 1.5 m

so, lambda/L = 1.1161

for closed closed string

lambda = 2L/n

1.1161 = 2/n

n = 1.7919 ( which is not an integer)

so standing wave mode cannot be set at 4 kg

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