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106.

Problem 16.37P (HRW)

A uniform spring whose unstretched length is L has a spring constant k. The spring is cut into two pieces of unstretched lengths and , with . We have to find (a) the corresponding spring constants and in terms of n and k. (b) If a block is attached to the original spring, it oscillates with frequency f. If the spring is replaced with the piece or , the corresponding frequency is or . We have to find and in terms of f.

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(a)

It is given that the unstretched length of the spring is L and that it has been cut into two pieces of lengths and , with . Let the spring of force constant k stretch by length x under a force F. It may be appreciated that at equilibrium each section of the spring will experience the same force F. We will use this property to answer the problem.

Let us say that the portion of the spring with unstretched length stretches by length when the uncut spring stretches by length x. And, the portion of the spring with unstretched length stretch by length .

We have

As each portion of the spring is subject to the same force F, we can now find spring constant, , for the portion with the length , and the spring constant, , for the portion with the length , by the requirement

(b)

If the original spring oscillates with frequency f when a block of mass m is attached to it, then

When the same block is attached to springs of lengths and , there will be SHM with frequencies determined by and , respectively. We have