# Difference between simple harmonic motion and angular SHM

I am not able to decipher when it is simple harmonic motion and when it is angular harmonic motion. Can we use both of them interchangeably? Can I know all the variable analogous for angular SHM (by which I mean use $$I$$ [moment of inertia] instead of mass and angle instead of displacement)?

Angular simple harmonic motion (SHM) occurs when a body's angular acceleration is proportional to its angular displacement from a fixed angular position and directed towards that position. If displaced and released, the body oscillates in angular fashion, that is the paths of each point are back and forward along arcs of circles centred on some fixed point. Another name for this sort of oscillation is a 'libration'.

The standard example used to be the balance wheel of a clockwork watch. It turned in one sense, slowed down, stopped, went faster and and faster in the other sense, slowed down, stopped and so on. The restoring torque on the balance wheel was provided by a flat spiral spring called a 'hair spring'.

If $$c$$ is the restoring torque provided by the spring over unit angle turned through by the balance wheel from its rest orientation, then, with the usual notation, $$c \theta=-I \ddot \theta.$$

A pendulum is another example, though it departs more and more from pure SHM the larger the maximum angle (angular amplitude).

You can indeed use moments of inertia, angular displacement, angular acceleration and torque instead of mass, linear displacement, linear acceleration and force, provided you make all these substitutions. It's arguably the neatest way to analyse pendulum motion.

If you're studying angular SHM, be sure not to confuse the angular frequency $$\omega$$ of SHM (equal to $$2 \pi / T$$) with the angular velocity ($$\Omega,$$ say) of the body itself (meaning $$\dot \theta$$ or $$d \theta/dt).$$

• No harm done. I can take a joke :-) – DanielSank Mar 18 '19 at 16:24