Question

In the figure below, a string, bed to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m. The separation L between P and Q is 1.80 m, and the frequency f of the oscillator Is fixed at 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 standing wave appears when the mass of the hanging block is 286.1 g or 643.7 g, but not for any intermediate mass. What is the linear density of the string? g/m

Answer 1

From the given Figure,

2 lambda = L

lamnda = 1.80/2 = 0.90 m

f = 120 Hz

v = lambda f = 0.90 x 120 = 108 m/s

v = sqrt[ T / linear density]

v^2 = T / mu

v1 / v2 = sqrt(643.7 / 286.1) = 1.50

v = lambda f

lambda1 / lambda2 = 1.50

lambd1 = L = 1.80 m

lambda2 = 1.2 m

v1 = lambda1 f = 216 m/s

v1^2 = T / mu

mu = (643.7 x 10^-3 x 9.8) / (216^2)

mu = 1.352 x 10^-4 kg/m