📂Physics

The Period of Simple Pendulum Motion is Independent of the Pendulum's Mass

Theorem

The period $T$ of a simple pendulum motion is independent of the mass of the pendulum $m$.

Description

Therefore, the period $T$ of a simple pendulum motion is independent of the pendulum’s mass, the amplitude’s size, etc., and depends solely on the pendulum’s length and the acceleration due to gravity.

Proof

The restoring force of the pendulum is as follows:

$$F=-mg\sin\theta$$

Since $x=l\theta$, when $\theta$ is sufficiently small, the following approximation holds:

$$\sin\theta \simeq \theta$$

At this time, the restoring force is:

$$\begin{align*} F =&\ -mg \sin\theta \\ =&\ -mg\theta \\ =&\ -mg\frac{x}{l} \\ =&\ -\frac{mg}{l} x \end{align*}$$

The restoring force of the pendulum can also be expressed as $F=-kx$. Therefore,

$$k=\dfrac{mg}{l} \quad \implies \quad \dfrac{m}{k}=\dfrac{l}{g}$$

The period is $T=\frac{2\pi}{w}=2\pi \sqrt{\frac{m}{k}}$, so

$$T=2\pi\sqrt{\frac{l}{g}}$$

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