Where:

We wish to find the equation of motion for this rigid body.

In analyzing pendulum physics a common simplification is to assume no friction at the pivot

Where:

The gravitational potential energy of the pendulum is:

where

Therefore,

The kinetic energy of the pendulum is:

where

Furthermore,

Applying the above equation for the conservation of energy for the pendulum we have,

Next, differentiate this energy equation with respect to time. This is a convenient way to obtain the equation of motion for the pendulum. We get,

This becomes:

A common simplification when analyzing pendulum physics is to assume that

Therefore,

This is a second order differential equation. It has the general solution:

where

In the above equation, the constant

A specific case is to assume that we have a simple pendulum, instead of a rigid body, as shown below.

In a simple pendulum the mass

For a simple pendulum the moment of inertia about point

and

The period of oscillation

This is the time it takes to complete one full cycle, or “swing”.

The frequency

To illustrate the motion of a pendulum, let's look at a specific case, where the amplitude

The motion shown in this graph is sometimes called

Return to

Return to

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