A string, tied to a sinusoidal oscillator at P and running over a support at Q,

Question

A string, tied to a sinusoidal oscillator at P and running over a support at Q, is stretched by a block of mass m, as shown in the figure. The separation L between P and Q is 1.2 m, and the frequency f of the oscillator is fixed at 50 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 99 g or 176 g, but not for any mass in between. Calculate the linear density mu of the string in g/m.

Explanation / Answer

here

the block is in equilibrium thus the tension in the string is

0 = F - mg

F = mg

to produce a standing wave in the string the mass of the block must satisfy

f = n * v / (2L) = (n/(2L) )* sqrt(m*g/u)

m = 4 * L^2 * f^2 * u / ( n^2 * g)

u = m*n^2 * g / (4 * L^2 * f^2)

then for n

n = 1 / (sqrt(m/m') - 1)

n = 1 / (sqrt(176/99)-1) = 3

then putting in the formula

u = 0.176 * 3^2 * 9.8 / (4 * 1.2^2 * 50^2)

u = 1.078 * 10^-3 kg /m

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