Question

Show that motion of bob of pendulum with small amplitude is linear $S.H.M$. Hence obtained an expression for its period. What are the factors on which its period depends?

Answer 1

To show that the motion of the bob of the simple pendulum is S.H.M

$1)$ Consider a simple pendulum of mass $m$ and length $L$

$L=l+r$

where $l=$ length of string and $r$ is the radius of the bob.

$2)$Let $OA$ be the initial position of pendulum and $OB$, be its instantaneous position when the string makes an angle $\theta$ with the vertical.

In displaced position, two forces are acting on the bob:

$a)$Gravitational force(weight)$mg$ in the downward direction.

$b)$Tension $T^{\prime }$ in the string.

$3)$Weight $mg$ can be resolved in to two rectangular components:

$a)$Radial component $mg\cos \theta$ along $OB$ and

$b)$Tangential component $mg\sin \theta$ perpendicular to $OB$ and directed towards mean position.

$4)mg\cos \theta$ is balanced by tension $T^{\prime }$ in the string,while $mg\sin \theta$ provides restoring force.

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

where negative sign shows that force and angular displacement are oppositely directed.Hence, restoring force is proportional to $\sin \theta$ instead of $\theta$.So, the resulting motion is not S.H.M.

$5)$If $\theta$ is very small then

$\sin \theta \approx \theta =\frac{x}{L}$

$\therefore F=-mg\frac{x}{L}$

$\therefore \frac{F}{m}=\frac{-gx}{L}$

$\therefore \frac{ma}{m}=\frac{-gx}{L}$

$\therefore a=\frac{-gx}{L}$ ....$\left(1\right)$

$\therefore a\propto -x$ since $\frac{g}{L}=$constant

Hence, the motion of the bob of a simple pendulum is simple harmonic.

Expression for the time period:

In S.H.M

$a=-{\omega }^{2}x$ ....$\left(2\right)$

Comparing equations $\left(1\right)$ and $\left(2\right)$

${\omega }^2=\frac{g}{L}$

But $\omega =\frac{2\pi }{T}$

${\left(\frac{2\pi }{T}\right)}^2=\frac{g}{L}$

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

$T=2\pi \sqrt{\frac{L}{g}}$ ....$\left(3\right)$

Equation $\left(3\right)$ represents the time period of the simple pendulum.Thus, the period of a simple pendulum depends on the length of the pendulum and acceleration due to gravity.