17.2 Simple Harmonic Motion

Simple Harmonic Motion

Definition: Simple Harmonic Motion

• Motion in which acceleration is directly proportional to displacement from a fixed point
• and acceleration is always in opposite direction to displacement.

Equation for Simple Harmonic Motion

$a=-{\omega }^{2}x$

Where,

• $a$: acceleration
• $x$: displacement
• $\omega$: angular frequency

Derivation: Acceleration

Important

• In Simple Harmonic Motion, frequency is constant, therefore angular frequency ($\omega$) is also constant.
• Negative sign shows that acceleration is in opposite direction of displacement.

Restoring Force

• The restoring force is the force responsible for SHM.
• It Always acts in the opposite direction of displacement and in same direction as acceleration.

Pendulum SHM Restoring Force

Showing that a given motion is Simple Harmonic

• To show that a given motion is Simple Harmonic Motion, the equation describing it should be able to be represented in the form $a=-kx$, where $k$ is a constant and equal to ${\omega }^2$.

• If given a $a-x$ graph, it should be a straight line passing through the origin with negative gradient.

What is not SHM?

• Any motion that does not obey the relation $a\propto -{\omega }^{2}x$ is not Simple Harmonic.

Example: Person jumping on trampoline

• A person jumping on a trampoline is not SHM.
• When the person is in the air, the only force acting is their weight, which is constant ($W=mg$).
• Because the restoring force is not proportional to displacement, it breaks the definition of SHM.

Not SHM Trampoline