Class 11 > Unit # 11:Oscillations > Simple Pendulum & S.H.M.

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Simple Pendulum, Proof of SHM & Time Period Derivation

Video Lecture: Simple Pendulum & SHM

Watch the complete step-by-step video lecture explaining the definition, mathematical proof, and time period derivation of a simple pendulum:

Definition & Proof that Simple Pendulum Motion is SHM

Definition: An ideal simple pendulum consists of a point mass (bob) suspended from a rigid, frictionless support by a light, inextensible string. When the bob is in its vertical position, the gravitational force $W$ acts vertically downward and the tension $T$ acts vertically upward, balancing each other ($W = T$).

When the bob is slightly displaced from its mean position, it begins to perform oscillatory motion under two forces:

1. The gravitational force $W = mg$ acting vertically downwards.
2. The tension $T$ acting along the string.

Resolving the weight $W$ into two rectangular components:

• Component along the length of the string: $W_{\parallel} = mg \cos\theta$
• Component perpendicular to the string (tangential): $W_{\perp} = mg \sin\theta$

Since there is no motion along the string, the net force along the string is zero, giving $T = mg \cos\theta$. The restoring force bringing the bob back to its mean position is provided by the perpendicular component:

$F_{\text{net}} = -mg \sin\theta$

According to Newton's Second Law of Motion ($F_{\text{net}} = ma$):

$ma = -mg \sin\theta \implies a = -g \sin\theta \quad \text{--- (i)}$

For small angular displacements ($\theta$), $\sin\theta \approx \theta$, so equation (i) becomes:

$a = -g\theta$

Since angular displacement is related to arc length ($S$) and radius ($r$) by $\theta = \frac{S}{r}$, and in this case $S = x$ (displacement) and $r = l$ (length of string):

$\theta = \frac{x}{l}$

Substituting $\theta$ into the acceleration expression:

$a = -g\left(\frac{x}{l}\right) \implies a = -\left(\frac{g}{l}\right)x$

Conclusion: Since $g$ and $l$ are constants, acceleration is directly proportional to displacement and directed towards the mean position ($a \propto -x$). Hence, it is strictly proved that the motion of a simple pendulum is Simple Harmonic Motion (S.H.M.).

Derivation of Expression for Time Period

According to the definition of time period $T$ for harmonic motion:

$T = \frac{2\pi}{\omega} \quad \text{--- (ii)}$

In general circular/harmonic motion, acceleration is related to angular velocity by:

$a = -\omega^2 x \quad \text{--- (iii)}$

From our pendulum derivation, the acceleration is:

$a = -\left(\frac{g}{l}\right)x \quad \text{--- (iv)}$

Comparing equations (iii) and (iv):

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

Substituting $\omega$ into equation (ii), we get the final expression for the time period of a simple pendulum:

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

Fig: Forces acting on the bob of a simple pendulum at displacement θ.

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