Comparison with Circular Motion

Idea: There is a very strong analogy between circular motion and simple harmonic motion. Consider a particle moving with constant angular velocity $\omega$ in a circle of radius R , as shown in Fig.13.2.

  

Figure 13.2: Analogue Between Circular and Simple Harmonic Motion

Its x-coordinate is given as a function of time by:

$$\omega$$ (9)

and the x-component of its tangential velocity is:

$$\omega \omega \omega$$ (10)

From this we deduce that

$$\omega^{2}_{} \omega^{2}_{}$$ (11)

which can be solved for v_x :

$$\sqrt{\omega^2(R^2-x^2)}$$ (12)

This is precisely the same as Eq.(13.7) relating the speed of an SHO to its position, providing we identify the radius with the amplitude and the angular velocity with $\sqrt{k/m}$ . Since the period (and frequency) are known for circular motion ( T = 2$\pi$/$\omega$ , f = 1/T ), this analogy allows us to deduce expressions for the period (and frequency) of the corresponding simple harmonic oscillator:

$${1\over f} \pi \sqrt{m\over k}$$ (13)

Similarly, the displacement and velocity as functions of time can also be deduced:

$$\sqrt{k\over m} \pi$$ x = (14)

$$\sqrt{k\over m} \sqrt{k\over m} \sqrt{k\over m} \pi$$ v = (15)

These expressions are plotted in the following figures.

  

Figure 13.3: Position and Velocity as Functions of Time

Note: ($\sqrt{k/m}$)t = 2$\pi$ft gives the argument of the sine and cosine functions in radians. Make sure that your calculator is set to radians when doing these problems.