## Mathematical Recreations

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#### Harmonic Oscillator With a Massive Spring (December 2001)

This problem involves some very basic physics. I've made an effort to spell it out, though, so don't worry if your physics is a bit rusty. In fact, this is a very standard introductory physics problem, which is always oversimplified (because they don't want the math to get in the way).

Consider a weight of mass M suspended by a spring with spring constant k, such that the force F exerted by the spring is given by

F  =  –k x.

This time, do not assume the spring is massless. The spring is uniform with mass m. For the moment, you can ignore air friction.

For those of you who could use a refresher on forces and motion, you might want to read the discussion of the simple harmonic oscillator. Here are some questions about the system:

• How much is the spring extended by the attached weight and its own weight?
• What is the lowest-order (slowest) frequency of oscillation for the system?
• What are the other frequencies of oscillation?

These are not too difficult. To really appreciate the fun, go out and get yourself a Slinky®. The knock-offs work just fine, but I recommend against the plastic ones. They just don't have enough mass. Find a balcony and have at it.

You should also look at the rotational modes of the system. Instead of making the Slinky® oscillate up and down, move the top end in a horizontal circle and find those modes of motion. You should find a low-order mode, where the slinky is aproximately straight; but you should also find higher-order modes with one or more nodes (where the slinky passes directly underneath your hand).

I expect that the mathematics of the rotating system are fascinating. The modes of rotation seem to be especially stable, since the slinky actually deforms to fit the mode. As a result, you can drive each mode with a range of frequencies. I'd love to get some feedback about the rotational system. I haven't tackled it yet.

Send all responses to .

Thanks,
Steve