### The Idea

This project explores an indirect way of measuring the gravitational acceleration of the Earth (or any other planet you may do this experiment on). Because gravitational acceleration affects how fast a pendulum swings, we can take advantage of that to find the gravitational acceleration provided by measuring two things: how *long* the pendulum is and how much *time* it takes to swing back and forth. (If you are less than eighteen years old, please be sure to get your parents' permission for any interplanetary travel for this project.)

### What You Need

- pendulum consisting of a mass supported by a string attached to a support
- pendulum with a long string and a large mass such as a bowling ball supported (safely and securely!) from the ceiling
- stopwatch

### Method

- Measure the period of a pendulum by measuring how long it takes for the pendulum mass to swing back and forth one time. Since this may be less than a second, more accurate measurements can be made by counting the time for 10 back-and-forth excursion, and then dividing by 10. Remember that in counting the cycles, the first cycle is counted when the mass returns to the point from which it was released and not at the point when it is first released.
- Measure the length of the pendulum. This is the distance in meters from the center of the hanging mass to the point of attachment.
- Try to minimize vibration of the ring stand or other support structure. Also keep the pendulum moving in two dimensions. Some people like to use a double string—one on either side of the mass—to keep the pendulum from wobbling.
- Calculate the gravitational acceleration, g (in meters per second per second), using the equation:

where *L* is string length in meters and *T* is time in seconds. (This will also work if you measure *L* in feet but you will get an answer in feet per second per second.) Try this multiple times and take the average to get the most accurate result. See Figure 22-1.

### Expected Results

Gravitational acceleration should be close to the accepted value of 9.81 m/s^{2}. Results within 2 percent of this value are easily achievable. If you are working with feet, gravitational acceleration is 32 feet/s^{2}. Longer pendulum lengths encounter less frictional loss and are easier to get an accurate period measurement. Remember, 100 centimeters is equal to 1 meter when determining the length of the pendulum.

The following set of values results in the 9.81 m/s^{2} target value for gravitational acceleration:

### Why It Works

It stands to reason that the greater the pull of gravity, the faster the pendulum motion and the shorter the period. This is given by the equation for the period of a pendulum:

where L is the length (in meters) and the period is one cycle back and forth (in seconds). Solving this for gravitational acceleration gives us:

The definition of the period of a pendulum is the number of seconds for it to swing *back and forth* one time.

### Other Things to Try

You have been captured by alien abductors and taken to an unknown planet (where there are no video games and no cable TV). You are able to remove your shoe, which has a 15 cm (0.15 m) shoelace. You find that when you let your shoe swing freely in a short arc, it returns to the point from which it was released in 1.26 seconds. To which planet should you direct the interplanetary rescue team? The gravitational acceleration on the various planets is: Venus 8.93 m/s^{2}, Earth 9.81 m/s^{2}, Mars 3.73 m/s^{2}, Jupiter 924.9 m/s^{2}, and Saturn 10.6 m/s^{2}. Try it. (Hint: the answer is this planet is one of Earth's neighbors in space possessing a very thin atmosphere, ice caps, and a reddish clay-like surface.)

### The Point

For a given string length, the period of the pendulum depends on the gravitational acceleration. This provides a fairly accurate method for measuring the local gravitational acceleration.

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