In the arrangement in the figure, a string, tied to a sinusoidal oscillator at P

Question

In the arrangement 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. The separation L between P and Q is 1.18 m, and the frequency f of the oscillator is fixed at 123 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 224 g or 350 g, but not for any intermediate mass. What is the linear density mu of the string?

Explanation / Answer

You need to form two equations from f=nV/2L
V=T/
2Lf/n=T/ where tension is 0.350gx9.8
and
2Lf/(n+1)=T/ where tension is 0.224x9.8

plug in numbers and solve them simultaneously for n
you'll notice you can substitute the n from the first one into the second equation and get the linear density.
2x123x1.18/n=(3.43/)
(290.28)/3.43=n
(290.28)/2.1952=n+1

((290.28)/2.1952)= ((290.28)/3.43 )+1

solve for

= 6.512981931 * 10^-4 kg/m

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