Question

How knowledgeable are you about the simple pendulum?

Answer 1

Definition:

1. A simple pendulum is a mechanical device that exhibits oscillating motion. In this arrangement, a point mass $"M"$ is suspended from fixed rigid support by a long thread which is inseparable from a string of length $"L"$
2. $F=-mg\sin \theta$ is the net force acting on m. It is a force that restores. $s=L\theta$ represents the distance away from the equilibrium position.

Simple Pendulum: Period and frequency relationship:

$$F_t=-mgsin\theta =\frac{m{d}^{2}s}{d{t}^2}$$,

Here $F_t=$Frequency, $m=$mass of the bob, $\theta =$angular of displacement,and $g=$acceleration due to gravity.

Now, in the case of small displacement where

$$sin\theta \equiv 0$$and along the arc is the motion of the bob, then

$s=L\theta$

Therefore,

$$\frac{d^2\theta }{d{t}^2}=-\frac{g}{L}sin\theta =-\frac{g}{L}\theta$$

or

$$\frac{d^2\theta }{d{t}^2}+\frac{g}{L}\theta =0$$

$$\frac{d^2\theta }{d{t}^2}+{\omega }_n^2\theta =0$$

Where, $${\omega }_n$$is given by,

$${\omega }_n=\sqrt{\frac{g}{L}}$$, here $L=$length of string.

and the period$\left(T\right)$ of motion is

$$T=\frac{2\pi }{{\omega }_n}=2\pi \sqrt{\frac{L}{g}}$$

$\theta =A\sin ({\omega }_{n}t+\varphi )$

This equation shows that the length of the string and the magnitude are$g$are the only variables that affect the period and frequency of a simple pendulum.

Hence, we need to know the string's length and its arc in order to calculate the pendulum's angle.