### Demos: 1S-02 Compound Pendulum

The pendulum oscillates about the axis

*X-X*in a plane perpendicular to

*X-X*. The weight

*W*is fixed, whereas the weight

*A*can slide up and down the shaft, altering the rotational inertia. The period of the pendulum about O is found from:

where

*d*is the distance to the center of gravity of

*A*and

*W*(assuming the other masses to be negligible). If

*A*and

*W*are also approximated as point masses, then

The center of gravity is located, relative to

*O*,

This leads to

Note that if

*M*is zero, we have the result for a simple pendulum. If

_{A}*M*and

_{A}= M_{W}*d*, the period of the pendulum is indefinitely large, as expected.

_{A}= d_{W}If the entire mechanism is tilted about the hinge through an angle q, the factor

*g*becomes

*gcosq*and the period will increase (Mach’s Pendulum).

**Directions:**For a qualitative approach to the demonstration, simply move the sliding mass to various positions and show the positional dependence. In particular, if the mass

*A*is placed such that it is located at the center of gravity, the period will become indefinitely large (infinite). If you wish to perform a quantitative analysis, careful measurements must be made of the distances. (You could measure

*g*if all other parameters are known.)

For the “Mach Pendulum,” leave the masses in position and lift the apparatus to some intermediate angle to show that the period diminishes with q.

**Suggestions for Presentation:**See

**Directions**above.

**Applications:**A variant of the Mach Pendulum was used in astronaut training years ago. The astronaut trainee was attached by harness to a long rope. He moved about on a ramp inclined at an angle such that the component of

*g*along the slope was 1/6 that of the Earth. In some ways, the astronaut was able to simulate movement on the moon.

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*Last Updated*: May 9, 2016 11:44 AM