The Simple Pendulum

If a pendulum of mass m attached to a string of length L is displaced by an angle $\theta$ from the vertical (see figure below),

  

Figure 13.4: The Simple Pendulum

it experiences a net restoring force due to gravity:

$$\theta$$ (16)

For small angles, sin $\theta$ $\approx$ $\theta$ , providing $\theta$ is expressed in radians (try it on your calculator for $\theta$ = 0.1,0.5,1.0 radians). In terms of radians,

$$\theta$$ = $${s \over L}$$ radians

where s is the arc length and L is the length of the string. Thus, for small displacements, s , the restoring force can be written:

F_r = - $$\left({mg\over L} \right)s$$.

Since the restoring force is proportional to the displacement, the pendulum is a simple harmonic oscillator with ``spring constant'' k = mg/L . The period of a simple pendulum is therefore:

 

$$\pi \sqrt{m\over k} \pi \sqrt{L\over g}$$ (17)

Note:

• In this small angle approximation, the amplitude of the pendulum has no effect on the period. This is what makes pendulums such good time keepers. As they inevitably lose energy due to frictional forces, their amplitude decreases, but the period remains constant.