A uniform cord of length ℓand mass m is hung vertically from a support. (a) Show that the speed

step 1 In this question, we have a uniform chord of length L and mass M that is hanging vertically. And

A uniform cord of length ℓand mass m is hung vertically from...

A uniform cord of length $\ell$ and mass $m$ is hung vertically from a support. $(a)$ Show that the speed of transverse waves in this cord is $\sqrt{g h}$ , where $h$ is the height above the lower
end. (b) How long does it take for a pulse to travel upward from one end to the other?

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Key Concepts

Transverse Wave Speed on a String

The speed of transverse waves on a string is determined by the tension in the string and its linear mass density. This is captured by the formula v = ?(T/?), showing that any variation in tension or mass density along the string will directly affect the wave’s speed. This principle is central in understanding wave propagation in strings and other flexible media.

Variable Tension in a Hanging Cord

In a hanging cord, the tension is not constant along its length because it is due to the weight of the material below any given point. This variable tension means that the wave speed varies along the cord depending on the height above the lower end, as the weight (and therefore the tension) increases progressively with height. Recognizing this variation is critical when analyzing wave motion in non-uniform tension systems.

Integration of Differential Time Elements

When the wave speed varies along the medium, calculating the total travel time of a pulse involves integrating the differential time, dt = dx/v(x), over the length of the medium. This method allows one to sum the contributions from each infinitesimal segment where the speed is different, which is a standard technique in problems involving variable-speed propagation.

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