# Measuring the Rate of Gravitational Acceleration of Various Ojects on the Earth

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### The Idea

Objects exposed to the force of gravity accelerate at the same rate. We proved that in the previous experiment. Here, we measure the rate of gravitational acceleration for all objects on the earth.

You measure acceleration two different ways in this experiment. In the first method, you use a stopwatch. We call this a ballpark experiment, which means we expect it to give a rough approximation rather than a very accurate result.

The second method involves the use of a motion sensor, which offers a greater degree of precision.

### Stopwatch method

• various objects: baseballs, golf balls, bowling balls, your physics textbook
• stopwatch
• tape measure

### Motion sensor method

• motion sensor with DataStudio software
• ring stand or other support to orient the motion sensor vertically, looking downward

### Stopwatch method

1. Use the tape measure to identify the distance the object will be dropped.
2. One person drops the object and the other person times the trip down.
3. Start the timer just as the object is released and stop it at the precise time it hits the ground. Try to avoid anticipating the release that will give too large a time measurement and an understated value for gravitational acceleration.
4. Calculate the gravitational acceleration using the equation g = 2d/t2, where d is the distance in meters and t is the time in seconds. Gravitational acceleration is measured in m/s2, which is read as meters per second squared or meters per second per second.

### Motion sensor method

1. Set up a motion detector mounted on a tablewith an unobstructed view of the floor, as shown in Figure 19-1.
2. Set up the motion detector to read distance versus time and velocity versus time. This can be accomplished by selecting the "velocity" file that comes with the DataStudio software package.
3. This measurement works best by increasing the frequency of the motion sensor measurement by increasing the sampling from 10 per second to 50 per second.
4. Align the motion sensor in the vertical direction.
5. Hold the ball just under the motion sensor, as shown in Figure 19-2. Start the readings and release the ball. Try to avoid imparting any vertical momentum to the ball by letting it drop without an initial push or delayed release.
6. Capture the motion of the ball through several bounces.
7. Measure the slope of the velocity versus time graph. Use either the initial descent or the first bounce. The initial descent has the advantage of having the largest statistical sample. The first bounce has the advantage of being free of errors associated with the release.

### Expected Results

For either method, the accepted value for gravitational acceleration is about 9.81 m/s2. This may vary slightly with location and elevation.

### Stopwatch method

For a typical outdoor high-school athletic bleacher about 15 feet above the ground (about 4.6 meters), an object will take about 1 second to fall. We learn in the next project that a person's reaction time can easily be as much as ¼ second. As a result, any given measurement may have an error of as much as about 25 percent. (This can be even greater because there can be nono-ffsetting errors for the start and stop time of the measurement.) This is not very precise, but it puts us in the ballpark. It is hard to improve on this because of the limitation in measuring time inherent in the use of a stopwatch. Some people find that listening for the ball to hit the ground is easier to time than trying to observe it visually. A greater distance to fall also reduces errors because the reaction time is a smaller percentage of the overall time measured.

The following chart summarizes expected times for various distances. Times measured in this range gives reasonable values for gravitational acceleration, g.

Another result expected is that, within the accuracy of this experiment, all objects fall at the same rate of acceleration, regardless of their mass.

Notice how sensitive the results are on the time measurement. For instance, suppose you drop a bowling ball from a 4.6 meter height and measure 1.1 seconds instead of 1.0 seconds. That 0.1 second error would result in a calculated value for gravitational acceleration of 7.6m/s2 instead of the expected value of 9.8 m/s2 or a 22 percent error. A 0.1 second error is less than the reaction time of most people so it is a good thing that we have another way to make this measurement.

### Motion sensor method

With a motion sensor, the range of measurements is much tighter. The position versus time graph is shown in Figure 19-3. Notice this shows a curved line typical of acceleration. As the ball falls, the position increases, s the portion of the curve sloping up to the right represents the falling motion. After the ball bounces off the floor, the distance increases, which generates the curved line that slopes down to the right. This graph shows an initial release and then two bounces. The data collection stops just before a third bounce.

A velocity versus time graph generated by a motion sensor is shown in Figure 19-4. Gravitational acceleration is given directly by the slope of the line. This can be determined by dividing the rise (change in velocity) by the run (corresponding change in time). The slope can also be found by using the slope tool located in the DataStudio pull-down menu. This graph shows the same drop followed by two bounces, as you saw in Figure 19-3. Notice the first bounce occurs just before 1.2 seconds. The ball reaches its first peak at 1.5 seconds and begins to fall again. In Figure 19-4, the velocity rapidly changes from positive (above the line) to negative (below the line).

Also notice one interesting aspect of the physics of free-fall, illustrated by Figure 19-4. After each bounce, the slope is the same below the zero line (bouncing up), at the zero line (at the highest point) and above the zero line (falling back down). What this means is gravitational acceleration is constant and affects an object in free-fall, regardless of whether it is moving up or down.

### Why It Works

Part 1 is a direct measurement and application of the basic motion formula:

a = 2d/t2

where we find the acceleration due to the force of gravity.

Part 2 measures the same thing, but it uses a much more precise measurement of the distance traveled in a given time. We know from Projects 1 and 2 that the slope of the distance versus the time graph gives a measure of velocity. Similarly, the slope of the velocity versus the time graph gives acceleration. Each bounce provides a replication of this experiment that can provide a separate data point.

### Other Things to Try

A motion sensor reveals the brief time that a ball encounters the ground as it compresses, decompresses, and eventually reverses direction. Some balls do this more quickly than others. This can be seen in time-lapse photography but can also be noticeable in the distance versus time graphs generated by motion sensor.

There is another method for measuring the Earth's gravitational acceleration using a pendulum. See Project 22. Compare this with the results you get with the motion sensor.

### The Point

This experiment gives two ways to measure the acceleration on any object caused by the gravitational force of the Earth. The first way is a direct measurement limited by the reaction time to record how long it takes an object to fall. The second method uses a motion sensor that captures this data with greater resolution and precision, and when interpreted graphically gives a more accurate value for gravitational acceleration. In either case, the correct value is 9.8 m/s2 or 32 ft/s2.