A simple pendulum consists of a mass on a string. It certainly looks like a simple harmonic motion system, although it moves back and forth along part of a circle rather than back and forth in a straight line.

The forces applied to the mass are the force of gravity and the tension in the string. The tension always points back to the point of rotation, so it produces no torque. Gravity, however, gives us a restoring torque. Applying Newton's second law for rotation:

Στ = Iα

-mgLsin(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 = -(mgL/I)θ

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

So, the angular frequency is ω = (mgL/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 + δ)

A physical pendulum is any pendulum where the mass is NOT all concentrated at one point. The general analysis we did for the simple pendulum applies down to the relationship:

So, the angular frequency is ω = (mgL/I)^½

Here L is not the length of the pendulum, but the distance from the point of rotation to the pendulum's center of gravity. The book uses h for this distance, so we should be consistent with that.

For a physical pendulum ω = (mgh/I)^½

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

θ(t) = θ_max cos(ωt + δ)