For procedures, see your mechanical laboratory manual.

Title: Simple Pendulum


Aim

To determine the acceleration due to gravity 'g'.

Theory


A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing.


A simple pendulum can be considered to be a point mass suspended from a string or rod of negligible mass. It is a resonant system with a single resonant frequency. A simple pendulum consists of a mass m hanging from a string of length L and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with the periodic motion.

For small amplitudes, the period of such a pendulum can be approximated by:

By applying Newton's second law for rotational systems, the equation of motion for the pendulum may be obtained

The motion of a simple pendulum is like simple harmonic motion in that the equation for the angular displacement is
which is the same form as the motion of a mass on a spring
The angular frequency of the motion is then given by 
compared to
 for a mass on a spring.
The frequency of the pendulum in Hz is given by
and the period of motion is then

The moment of inertia can be measured using a simple pendulum because it is the resistance to the rotation caused by gravity.

References

  1. Moment of inertia, wikipedia.org.
  2. Simple Pendulum, Hyperphysics, Georgia  State University.
  3. Pendulum, wikipedia.org.
  4. Vibrations and Waves - Lesson 0 - Vibrations, physicsclassroom.com.
  5. The Simple Pendulum, The Pennsylvania State University.
  6. S. Rajasekaran, in Structural Dynamics of Earthquake Engineering, 2009