16.6 Uniform circular motion and simple harmonic motion  (Page 2/2)

Now let us use [link] to do some further analysis of uniform circular motion as it relates to simple harmonic motion. The triangle formed by the velocities in the figure and the triangle formed by the displacements ( $X,x,$ and $\sqrt{X^2-x^2}$ ) are similar right triangles. Taking ratios of similar sides, we see that

We can solve this equation for the speed $v$ or

This expression for the speed of a simple harmonic oscillator is exactly the same as the equation obtained from conservation of energy considerations in Energy and the Simple Harmonic Oscillator . You can begin to see that it is possible to get all of the characteristics of simple harmonic motion from an analysis of the projection of uniform circular motion.

Finally, let us consider the period $T$ of the motion of the projection. This period is the time it takes the point P to complete one revolution. That time is the circumference of the circle $\mathrm{2\pi }X$ divided by the velocity around the circle, $v_{\text{max}}$ . Thus, the period $T$ is

We know from conservation of energy considerations that

Solving this equation for $X/v_{\text{max}}$ gives

Substituting this expression into the equation for $T$ yields

Thus, the period of the motion is the same as for a simple harmonic oscillator. We have determined the period for any simple harmonic oscillator using the relationship between uniform circular motion and simple harmonic motion.

Some modules occasionally refer to the connection between uniform circular motion and simple harmonic motion. Moreover, if you carry your study of physics and its applications to greater depths, you will find this relationship useful. It can, for example, help to analyze how waves add when they are superimposed.

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Section summary

A projection of uniform circular motion undergoes simple harmonic oscillation.

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