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Chapter 14: Problem 5

A heavy cable is hanging vertically, its bottom end free. How will the speed of transverse waves near the top and bottom of the cable compare? Why?

Short Answer

Expert verified
The speed of transverse waves at the top of the cable is greater than at the bottom due to higher tension at the top of the cable.

Step by step solution

01

Understanding the system

Firstly, consider the tension in the cable. Tension is maximum at the top because the cable's weight hanging below exerts a downward force. As we progress towards the bottom, the tension decreases because the amount of cable (and hence weight) hanging beneath reduces.
02

Applying Physics of Waves

The speed \(v\) of a wave travelling along a stretched string or cable is given by \(v = \sqrt{\frac{T}{\mu}}\) where \(T\) is the tension in the cable and \(\mu\) is the linear mass density of the cable. Notice from this equation that wave speed increases with increasing tension and decreases with increasing mass density. Therefore, since the tension is greatest at the top of the cable and lowest at the bottom, the speed of the wave will be greatest at the top and slowest at the bottom.
03

Comparing wave speed

Given that the linear mass density remains constant throughout the length of the cable, the variation in wave speed is solely due to the changing tension. Therefore, the speed of transverse waves at the top of the cable is greater than the speed at the bottom due to higher tension at the top.

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