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Chapter 6: Problem 6

A person attaches a paper clip to each coil of a Slinky-about 90 in all-in such a way that waves will still travel on it. What effect, if any, will the paper clips have on the speed of the waves on the Slinky?

Short Answer

Expert verified
Answer: The speed of the waves traveling through the Slinky will decrease due to the increase in mass per unit length caused by attaching paper clips to each coil.

Step by step solution

01

Understand the wave speed formula

The wave speed is determined by the properties of the medium through which the wave travels. For a Slinky, we can use the formula \(v = \sqrt{\frac{T}{\mu}}\), where \(v\) is the wave speed, \(T\) is the tension in the spring, and \(\mu\) is the mass per unit length.
02

Determine the effect of paper clips on the mass per unit length

Attaching paper clips to each coil will increase the mass per unit length (\(\mu\)) of the Slinky. Let's consider the mass of each paper clip as \(m_p\). Since there are 90 paper clips attached to the Slinky and each coil has one paper clip, the total mass of the paper clips will be \(90m_p\). If the original mass per unit length of the Slinky is \(\mu_0\), then the new mass per unit length will be \(\mu_1 = \mu_0 + \frac{90m_p}{L}\), where \(L\) is the length of the Slinky.
03

Analyze the effect on wave speed due to the change in mass per unit length

Now, with the increase in mass per unit length (\(\mu_1\)), the wave speed can be calculated using the same formula \(v_1 = \sqrt{\frac{T}{\mu_1}}\). Since tension (\(T\)) remains constant, an increase in the mass per unit length (\(\mu_1\)) will result in a decrease in the wave speed (\(v_1\)).
04

Conclusion

Attaching paper clips to each coil of the Slinky will increase the mass per unit length (\(\mu\)), which will ultimately cause the speed of the waves traveling through the Slinky to decrease.

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