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It is known that in simple harmonic motion, the total energy of the system is proportional the square of the amplitude, but how can I prove that for a simple pendulum where amplitude is the arc length of the part of the circle? Much obliged for any help.

cwwwonggg
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1 Answers1

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Suppose the maximum angle of the pendulum to the vertical direction is $\theta$, where the pendulum has a zero-velocity. According to the conservation of mechanical energy, the total energy of the system, $E$, should be:

$$E=mgl(1-\cos\theta )$$
If $\theta$ is small enough, we can expand $\cos\theta$ up to the second order:

$$\cos(\theta)\approx 1-\frac{\theta^2}{2}$$

Plug this into the energy formula, we can get:

$$E=mgl(1-\cos(\theta))\approx mgl\left[1-\left(1-\frac{\theta^{2}}{2}\right)\right]=\frac{mgl\theta^2}{2}$$

The total energy of the simple pendulum system is proportional to $\theta^2$. It should be noted that this relation just holds when $\theta$ is very small, only based on which $1-\frac{\theta^2}{2}$ can be a good approximation of $\cos\theta\; .$

bitsoal
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