Another simple harmonic motion system is a pendulum. A simple pendulum consists of a mass on a string.

The forces applied to the mass are the force of gravity and the tension in the string. A component of the force of gravity provides the restoring torque. Applying Newton's second law for rotation:

Στ = Iα

-mg L sin(θ) = Iα

The negative sign is because the torque is opposite to the angular displacement.

For a system to undergo simple harmonic motion we must have the acceleration proportional to the negative of the displacement. We almost have that here. For small angles, however, we can use the small-angle approximation:

sin(θ) = θ

This gives, at small angles:

-mgLθ = Iα

α

= -

m g L I

θ

A hallmark of simple harmonic motion is that α = -ω ²θ

So, the angular frequency is   ω

=

(

m g L I

)

½

For a simple pendulum the rotational inertia is given by:

I = mL²

This gives   ω

=

(

g L

)

½

Note that this is independent of the mass of the pendulum.

The general equation giving the position of the pendulum as a function of time is:

θ(t) = θ_max cos(ωt + φ)

A physical pendulum is any pendulum where the mass is NOT all concentrated at one point.

For a physical pendulum   ω

=

(

m g h I

)

½

Here h is not the length of the pendulum, but the distance from the point of rotation to the pendulum's center of gravity.

If the physical pendulum is a rod of length L supported at one end:

h

=

L 2

and

I

=

1 3

mL2

This would give an angular frequency of:

ω

=

(

3g 2L

)

½

This has the same frequency as a simple pendulum with a length of 2L/3.