# Pendulum Motion

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Swinging back and forth, the solo trapeze is a giant pendulum, just like the one in a grandfather clock. The time it takes to swing forward, then back to where it started is called the period. Surprisingly, this time has very little to do with the height of the swing, or amplitude. It depends mainly on the length of the pendulum, the longer the pendulum, the longer the period. In this unit, you'll learn how to find how to find a pendulum's period from its length, and vice-versa.

## Watch the Video

## Circus Physics: Pendulum Motion

Watch Regina swing gracefully through the air on the solo trapeze. Find out what this circus act has in common with a grandfather clock.

### Questions to Consider While Watching the Video

- How is the trapeze like a grandfather clock?
- What influences how long it takes to go back and forth?
- What would happen if Regina were lighter or heavier?

## Digging Deeper

Here's a diagram showing Regina on the trapeze in three different positions: sitting, standing, and hanging below.

The length of the pendulum, *L*_{1}, *L*_{2}, or *L*_{3} is always the distance from the pivot point near the ceiling to Regina's center of mass, m. Notice that *L*_{3}—when Regina is hanging on the rope below the trapeze bar—is the longest, followed by *L*_{1}—sitting on the bar. *L*_{2}—standing on the bar—is the shortest. Which of these positions, if held for the entire swing, would take the longest time to go back and forth?

Surprisingly, the period has very little to do with how high Regina swings. This is because for higher swings, the distance to swing through may be longer, but the speed Regina has is also greater. For lower swings the distance is shorter, but her speed is slower. This means that the equation to find the period of the swing...

*T* = 2pi*√(l/g)

...depends only on the length of the swing, *l*, and the acceleration due to gravity, *g*.

We can see from the period equation that longer l's will result in longer periods. This means that when the length is *L*_{3}—when Regina is hanging on the rope below the trapeze bar—the period of the swing will be the longest.

### Clocks

We can use the same equation to figure out how long the pendulum in a grandfather clock should be if it is to swing with a period of two seconds, one second for "tick" and one for "tock."

We set T = 2s and then solve the equation for l, giving:

*l* = *g**(*T*/2pi)²

*l* = *g**(2 sec/2pi)²

*l *= 0.99 m

## Your Turn

Use the concepts and formulas from this unit to figure out the following:

A monkey swings across a stream on a vine anchored to a branch five meters away, centered over the middle of the stream. How long does it take him to swing to the other side?

##### Answer: ≈2.24 seconds

Use the pendulum period equation, *T* = 2pi*√(l/g), with* l* = 5 meters and *g* = 9.8 m/s. This gives:

*T* = 2pi*√(5m/9.8m/s/s) ≈ 4.49 s

This is the time to go back and forth, however, so divide it by two to find the time to swing one way:

4.49 s / 2 ≈2.24 seconds

## Further Exploration

If you'd like to learn more about the trapeze and pendulum motion, check out these links: