1.11 Simple Harmonic Motion (Simple Pendulum):

A simple pendulum is a heavy point mass suspended by an inextensible, weightless string from a rigid support. The pendulum oscillates in a vertical plane about the point of suspension as shown in the figure below.

Let $m$ be the mass of small bob is suspended by a thread from the rigid support $XY$ as shown in the figure.

Let $A$ be the equilibrium position of the pendulum. If the bob is displaced $y$ from the equilibrium position by an angle $\theta$ to new position $B$ and then released, it oscillates simple harmonically about its equilibrium position over a fixed path $CAB$.

At position B, The bob is under the action of following forces,

i) The component $mg\;Cos\theta$ of its weight $(mg)$ which balances the tension $(T)$ on the string,

i.e. $mg \; Cos\theta = T$ .......... (i)

ii) The component $mg\;Sin\theta$ of its weight $(mg)$ which provides the restoring force,

i.e. $F = -\; mg \; Sin\theta$ .......... (ii) 

[$-\;ve$ sign indicates that restoring force acts opposite to displacement]

If $a$ is the acceleration of the bob towards mean position due to restoring force, then

$F = ma$ .......... (iii)

From equation $(ii)$ and $(iii)$, we get

⇒ $a = -\;g \; Sin\theta$ .......... (iv)

If $\theta \; ￫ \; 0$, the motion of the bob along the curve $CAB$ considered as straight line. In such condition,

$Sin\theta ≈ \theta ≈$ $ \frac{arc \; AB}{l} = \frac{y}{l}$ [∵  $arc \; AB = AB = BN = y$ (Linear displacement)]

or, $a = - \; g$ $ \frac{y}{l}$

or, $a = -$ $ \frac{g}{l}$$y$ .......... (iv)

Since, $\frac{g}{l}$ is a constant for a given pendulum at a given place.

$a ∝ -\; y$ .......... (v)

Since acceleration is directly proportional to the linear displacement from mean position. This shows that the motion of the simple pendulum is simple harmonic.

Further, for the Time Period of simple pendulum,

The acceleration of the any particle in SHM is given by,

$a = - \; \omega^2\; y$ .......... (vi)

Comparing equation $(v)$ and $(vi)$, we get

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

$\omega = $ $\sqrt{\frac{g}{l}}$

$\frac{2 \pi}{T}$ $ =  $ $ \sqrt{\frac{g}{l}}$                           [∵ $\omega = \frac{2\pi}{T}$]

$T = 2\pi$ $ \sqrt{\frac{l}{g}}$ .......... (vii)

Thus, the time period of the simple pendulum is entirely depends on $l$ and acceleration due to gravity  $g$ at a place.

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E 1.11 Periodic Motion (Basic Introduction):

"Back and forth Motion"

The motion which the particle repeat the similar condition after a fixed interval of time is called periodic motion.

Key terms of Periodic Motion:

» Oscillation: Repeated back and forth motion over the same path about an equilibrium position.

» Restoring Force: When a body is displaced from its equilibrium position, the force tends to restore it in a equilibrium position. This force is called the restoring force.

» Simple Harmonic Motion: Repetitive 'to and fro' or 'up and down' motion from mean (or equilibrium) position. So the maximum displacement of one side from the mean position is equal to the maximum displacement of other side from the mean position.

» Amplitude: The maximum magnitude of displacement from equilibrium position. It is always positive, denoted by $A$.

» Period: The time to complete one cycle, denoted by $T$. It is always positive.

» Frequency: The number of cycles per unit time, denoted by $f$.

» Angular Frequency: The angular frequency $\omega$ is $2\pi$ times the frequency.

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