Help please?

Question

One end of a horizontal string is tied to a wall, and the other end is tied to an object with weight W that hangs over a pulley to hold the string taut. The object is large enough that the string never moves at the pulley. Under these conditions, the string vibrates with wavelength and frequency in its first harmonic.

a) If we add enough weight to double W without appreciably stretching the string what will be the wavelength and frequency of the string's first harmonic vibration?

b) If we don't change W, but move the pulley so that the vibrating part of the string is half as long, what will be the wavelength and frequency of the string in its first harmonic?


c) If we now double W (without appreciably stretching the string) and at the same time move the pulley so that the vibrating part of the string is twice as long as it originally was, what are the wavelength and frequency of the string in its first harmonic?

Answer 1

I can't give you actual frequencies because there isn't enough information, but the question really concerns frequency change from a reference value.
For transverse vibration of a tensioned wire or string,
frequency F = sqrt(T/(m/L))/(2*L) (see ref.) where T=tension (=weight W), L=length, m=mass. Since m/L is mass of a unit length of string, it is constant when you are only varying T and L, so we can say F is proportional to sqrt(T)/L.
So in (a), doubling T increases F by sqrt(2).
In (b), multiplying L by 0.5 doubles the frequency.
In (c), doubling both T and L multiplies frequency by sqrt(2)/2.