explain steps to get the answer A rope of total mass m and length L is suspended

Question

explain steps to get the answer

A rope of total mass m and length L is suspended vertically. Analysis shows that for short transverse pulses, the waves above a short distance from the free end of the rope can be represented to a good approximation by the linear wave equation discussed in Section 16.6. Show that a transverse pulse travels the length of the rope in a time interval that is given approximately by delta = 2 L/g. Suggestion: First find an expression for the wave speed at any point a distance a* from the lower end by considering the rope's tension as resulting from the weight of the segment below that point.

Explanation / Answer

We know that v=(T/) this is equation 1

For a string hanging vertically, mass of the rope below a point, x, on the string (measured from the bottom), is simply m = x* = x*M/L

F=mg=x**g

This force F is nothing but equal to tension in the string

T=x**g

Substituting in equation 1

v=(T/)

v=(x**g/)

v=(xg)

v=(xg)1/2

this is the expression of wavelocityve speed at any pont x distance from the lower end

The speed, v(x), is dx/dt, so we have that:

dx/dt = (g*x)1/2

This is a separable equation:

(g*x)-1/2dx = dt

Integrating the left hand side from x = 0 to x = L, and the right hand side from t = 0 to t = T (where T is the total travel time) we have:

T=(2/g) *[L-0]

T=2*(L/g)

GOOD LUCK!!!

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