Acceleration of Gravity
If Galileo was correct and all objects accelerate towards the earth at the same rate, then the value of that acceleration will be an important number to measure.
There are a number of ways to determine the acceleration due to gravity:
I. The most direct method would be to drop an object and to record its motion for later measurement:
II. Often the acceleration of gravity can be determined more precisely from the swing of a pendulum. A pendulum swings in the arc of a circle. When the mass is away from its center position, a portion of the downward pull of gravity is in the direction towards the center position. This restoring force depends on the angle away from vertical and the strength of the gravitational force. The time of the swing depends on the changing speed and swing distance. For small angles of deflection, combining the equations results in the period, T, (time of a round trip) depending only on the length of the pendulum and the acceleration due to gravity:
- Choose an object which has considerable density (more precisely, considerable mass and relatively small surface area) so that the wind resistance during its fall will be insignificant.
- Arrange a camera to record a video sequence the fall (including its initial motion) over a known distance.
- Analyze the recorded motion counting the number of captured images in the motion sequence to determine the elapsed time of fall. (Standard video records 30 frames/second but your camera may record images at another rate. Varify your method of measuring time against a known time interval.)
- According to Galileo, the distance fallen from rest, d = 1/2 at2. Solving for
a we get a = 2d / t2. Substitute the distance fallen from rest and elapsed time to determine the acceleration due to gravity.
- But there is an inherent flaw in capturing images separated at finite time intervals. The fall may have started at a time between images, allowing the elapsed time to be long by up to the time between images. It might be more accurate to calculate a speed near the start of the fall but after the first falling interval, vi, and a later speed near the end of the sequence, vf, and use Galileo's definition of acceleration, a ≡ vf - vi / t.
- Using a long, thin string, hang a large mass from a very stationary support.
- Usings swings of small angles, time a large number (~100) round trip swings to determine the average period of the pendulum, T.
- Measure the length of the pendulum from the center of the mass to the stationary support. You may use any convenient unit of length, but using standard units will result in more useful values. (The meter is the universally accepted unit of length.)
- Substitute the pendulum's length, l, and its period, T, into the equation and solve to find
Finally, record your procedures, measurements, and findings in your journal. If you need course credit, use your observations recorded in your journal to construct a formal report