Question

Is a pendulum a simple harmonic motion?

Answer 1

Explanation:

1. In actuality, a light string whose weight is minuscule in comparison to the weight of the bob can be viewed as a weightless fiber, and a small but hefty spherical can be regarded as a point mass.
2. Suppose that a simple pendulum of length $L$ is displaced through a small angle $\theta$ and released.
3. It oscillates two sides of its equilibrium position.
4. At the displaced position, forces act on the bob are$\left(1\right)$its weight $mg$ and$\left(2\right)$ the tension $T$in the string.
5. Resolved $mg$ into two components $\mathrm{mgsin}\left(\mathrm{\theta }\right)$ perpendicular and $\mathrm{mgcos}\left(\mathrm{\theta }\right)$ parallel to the string.
6. The component $\mathrm{mgcos}\left(\mathrm{\theta }\right)$ is balanced by the tension in the string.
7. The component $\mathrm{mgsin}\left(\mathrm{\theta }\right)$ is unbalanced. This acts as restoring force.
8. $\mathrm{F}=-\mathrm{mgsin}\left(\mathrm{\theta }\right)$ where the negative sign indicates that force is the opposite.
9. But $\left(\theta \right)$ is very small, $\sin \left(\mathrm{\theta }\right)=\mathrm{\theta }$

Step 1: Determine the pendulum's value in relation to the simple harmonic motion.

$$\begin{gathered} \Rightarrow \mathrm{F}=-\mathrm{mg\theta } \\ \Rightarrow \mathrm{F}=-\frac{\mathrm{mgX}}{\mathrm{L}}\left(\Rightarrow \mathrm{\theta }=\frac{\mathrm{X}}{\mathrm{L}}\right) \\ \Rightarrow \mathrm{F}=-\left(\frac{\mathrm{mg}}{\mathrm{L}}\right)\mathrm{X} \end{gathered}$$

Step 2: Further solution to the above expression

Put,$F=ma$

$$\begin{gathered} \Rightarrow \mathrm{ma}=-\left(\frac{\mathrm{mg}}{\mathrm{L}}\right)\mathrm{X} \\ \Rightarrow \mathrm{a}=-\left(\frac{\mathrm{g}}{\mathrm{L}}\right)\mathrm{X} \end{gathered}$$

The linear S.H.M. motion of the basic pendulum.

$\Rightarrow \mathrm{a}=-\left(\frac{\mathrm{g}}{\mathrm{L}}\right)\mathrm{X}$

Step 3: Condition for simple pendulum:

1. Bob must be a little spherical that weighs a lot.
2. It needs to be supported by a thin spring.
3. It needs to have firm support.
4. There must be very little amplitude.

Consequently, a simple pendulum exhibits simple harmonic motion in some circumstances.