17.3 SHM from Circular Motion

Linking SHM to Circular Motion

Comparing Simple Harmonic Motion to Circular Motion

• Imagine a disc rotating in a horizontal plane. A vertical handle is attached to the edge of the disc.

  

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• A beam of light is shone horizontally at the disc. The shadow of the disc and the handle is seen on a screen.
• The shadow of the handle on the screen will move back and forward with Simple Harmonic Motion.

  

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Displacement

Derivation of Displacement ( $x$)

• The displacement $x$ of the shadow on the screen is equal to the distance $DC$.
• From the right angled triangle formed by the radius $r$ and angle $\theta$: $x=DC=r\sin \theta$
• By definition, angular velocity $\omega$ is the rate of change of angular displacement $\theta$: $\omega =\frac{\theta }{t}⟹\theta =\omega t$
• Substituting $\theta =\omega t$ into $x=r\sin \theta$: $x=r\sin \omega t$
• Since the radius of the circle $r$ represents the maximum displacement from the equilibrium position, it is the amplitude $x_0$: $x=x_0\sin \omega t$