A uniform rope with mass m and length l hangs vertically from the ceiling. a) De

Question

A uniform rope with mass m and length l hangs vertically from the ceiling. a) Determine the tension at a height y from the bottom of the rope. Hint: The tension at any point is equal to the weight of the part of the rope below that point. b) Determine the speed of a transverse wave at height y on the rope. c) Determine the time it takes for a wave to travel the length of the rope. Hint: The time is t dt = . Use the fact that the speed is v dy dt = / and substitute in order to get the integral in terms of y and dy, with limits from 0 to l.

Explanation / Answer

A)

Let T(y) be the tension as a function of height. Consider a small piece of the rope between y and y + dy (0 y L). The forces on this piece are T(y + dy) upward, T(y) downward, and the weight g dy downward.
Since the rope is at rest, we have T(y + dy) = T(y) + g dy.
Expanding this to first order in dy gives T0(y) = g.
The tension in the bottom of the rope is zero, so integrating from y = 0 up to a position y gives
T(y) = gy.
As a double-check, at the top end we have T(L) = gL, which is the weight of the entire rope, as it should be.
Alternatively, you can simply write down the answer, T(y) = gy, by noting that the tension at a given point in the rope is what supports the weight of all the rope below it.

b)

The wave speed at any point on the rope is given by v = sqrt (T/u) , where T is the tension at that point and u is the linear mass density.
Because the rope is hanging the tension varies from point to point.

Consider a point on the rope a distance y from the bottom end.
The forces acting on it are the weight of the rope below it, pulling down, and the tension, pulling up. Since the rope is in equilibrium, these forces balance. The weight of the rope below is given by u*g*y, so the tension is T = u*g*y.
Hence the wave speed is v = sqrt (ugy/u) = sqrt(gy).

c)

The time dt for the wave to move past a length dy, a distance y from the bottom end,
is dt = dy / v = dy/sqrt(gy) and the total time for the wave to move the entire length of the rope is
t = 0 to L dy/sqrt(gy)
= 2 (sqrt ( L / g))

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