Hooke’s Law expresses the dynamics of simple harmonic motion where the restoring force is directly proportional to the displacement from equilibrium.

A harmonic oscillator’s acceleration is given as follows

Where is a proportionality constant.
- Not all periodic motion is harmonic. However, for small enough , the motion is approximately harmonic
Simple Harmonic Motion is the projection of uniform circular motion onto a diameter. We refer to the analogous circle as the reference circle.
The angular speed of a reference circle is the angular frequency of the simple harmonic motion
The period and frequency in Simple Harmonic Motion are independent of Amplitude
The displacement can be calculated as a function of time

Where is the phase angle, that is the initial angle in the reference circle. It is the point in the cycle the motion was at time .
- Taking the derivative of the above gives the velocity and acceleration
The total energy present in simple harmonic motion is given by
- Observe that when the object is displaced by its Amplitude, the Potential Energy is at its maximum. The Kinetic Energy is since the restoring force pulls the object back to equilibrium and at the amplitude (the farthest it goes), the object is at rest.
- Conversely, at the equilibrium point, the velocity is at a maximum and the displacement point from the equilibrium point is . The Potential Energy is .

Angular Simple Harmonic Motion

Angular Simple Harmonic Motion involves restoring torques proportional to an angular displacement from the equilibrium position.
We have an analogue for Hooke’s Law where is the torsion constant
The angular frequency and frequency are obtained by using the rotational analogues to mass The angular displacement is obtained similarly

Simple Pendulum Motion

A simple pendulum is an idealized model consisting of a point mass suspended by a massless, unstretchable string.
The path of the point mass is an arc with radius , the length of the string

The tangential component of the force on the mass is given by

We can use a first order approximation ¹
The angular frequency is given by

The period of the pendulum is independent of the mass of the bob. It is only dependent on the length of the string
Suppose the pendulum is not a point mass. Let be the pivot point (i.e., the axis of rotation) The center of gravity is units away from . The weight of the object causes a restoring torque

The torque is clockwise when the displacement is counterclockwise.
- Using a first order approximation of we have
- The above lets us calculate the moment of inertia given the object’s mass and its behavior when allowed to oscillate and the distance to an axis of rotation.