2) Assuming that the wave on a string is represented by y(x, t) = yosin[2pi/lamb

Question

2) Assuming that the wave on a string is represented by y(x, t) = yosin[2pi/lambda (vt-x)] where y is the transverse displacement at time t (sec) of the piece of string at x (m). yo, v and 'lambda' are all constants. Find the velocity and acceleration of the small piece of string at x = 10 m as a function of time. Making use of the fact that the piece of string satisfies Newton's Second Law, show that the piece of string is acted on by a Hooke's Law force. (i.e. a force that can be written in the same form as the restoring force from an ideal spring)

Explanation / Answer

v = dy/dt = y0*2*pi*v/lambda cos( 2 pi/lambda ( vt -x) )

at x = 10

v = y0*2*pi*v/lambda cos( 2 pi/lambda ( vt -10) ]

a = dv/dt = - y0*(2*pi*v/lambda)^2 *sin(2 pi/lambda (vt -x) )

at x = 10

a = - y0*(2*pi*v/lambda)^2 *sin(2 pi/lambda (vt -10) )

F = m a = - m y0*(2*pi*v/lambda)^2 *sin(2 pi/lambda (vt -x) )

but y = y0**sin(2 pi/lambda (vt -x)

F = - m ( 2 pi v/lambda)^2 y

so hooke's law with k = m (2 pi v/lambda)^2

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