Find the transverse velocity of the point

Question

A certain string has a linear mass density of 0.25 kg/m and is stretched with a tension of 25 N. One end is given a sinusoidal motion with frequency 5 Hz and amplitude 0.01m. At time t=0, the end has zero displacement and is moving in the positive y-direction. (v=10 m/s, A=0.01m, ω=31.4 rad/s, T=0.2 s, λ=2.0 m, k=3.1 4/m)

 

Find the transverse velocity of the point, x=0.25 m at time t=0.1s

Answer 1

It depends how your class is looking at this problem but I will jump to the displacement equation for a standing wave. This gives the displacement up and down of a string under tension.

y (x,t) = A cos(2π/λ x) sin(2 π f t)

or

y (x,t) = A cos(2π/λ x) cos(2 π f t)

I am putting the oscillating end at x=0 m but there is some ambiguity. For the condition at t=0 only the first equation will work.

The λ in the equation is the wavelength of the wave and the f is the frequency. There are several different versions of this equation with variables exchanged. The problem gave values of variables connected though other equations and they match up to three significant figures.

The transverse velocity would be the derivative of displacement.

V_y (x,t) = A (2 π f ) cos(2π/λ x) cos(2 π f t)

V_y(0.25 m, 0.1 s) = 0.1 m (2 π 5 s) cos( 2 π / 2.0 m 0.25 m) cos(2 π 5 s 0.1 s) = -π √2 m/s ≈ -4.44 m/s

Also double check my algebra.