Pendulum Motion

Overview

A pendulum is one of the most important oscillating systems in physics. It consists of a bob suspended from a fixed point so that it can swing freely under the influence of gravity.

For small angular displacements, a simple pendulum executes motion that closely approximates SHM. Because of its regular period, the pendulum has historically been used in clocks, timing devices, and experiments to determine gravitational field strength.

Definition

A simple pendulum is an idealised model consisting of:

• a point-mass bob of mass $m$;
• a light inextensible string of length $l$;
• a fixed frictionless pivot;
• motion under gravity with negligible air resistance.

The equilibrium position is the lowest point of the swing, where the string is vertical.

Why It Matters

Pendulum motion links mechanics, oscillations, circular motion, energy methods, and practical measurement. It is also a standard example showing how an approximate SHM model arises from a physical force law.

The pendulum formula is useful but conditional: it requires small angular displacement.

Key Representations

Restoring Force

When displaced by angle $\theta$, weight $m\vec{g}$ acts downward. Resolving weight gives a signed tangential restoring-force component:

$$F_t=-mg\sin \theta$$

The negative sign indicates that this tangential component acts toward equilibrium after a positive angular displacement is chosen. Here $F_t$ is a signed tangential component along the chosen arc direction, not a full vector equation.

Why the Pendulum Oscillates

If displaced:

1. restoring force accelerates the bob toward equilibrium;
2. the bob gains speed;
3. inertia carries it past equilibrium;
4. restoring force reverses direction;
5. the cycle repeats.

Thus oscillation results from gravity providing restoring force and inertia carrying motion through equilibrium.

Small-Angle Approximation

For small angular displacements in radians:

$$\sin \theta \approx \theta$$

This is typically good for angles below about $10^{\circ }$ and often acceptable up to about $15^{\circ }$, depending on required precision.

Then:

$$F_t\approx -mg\theta$$

Derivation of SHM

Arc displacement along the tangent is:

$$x=l\theta$$

so:

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

Using signed tangential components, $F_t=m{a}_t$:

$$m{a}_t=-mg\theta$$

Substitute $\theta =x/l$:

$$m{a}_t=-mg\frac{x}{l}$$

Therefore:

$$a_t=-\frac{g}{l}x$$

Comparing with:

$$a=-{\omega }^{2}x$$

gives:

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

Hence pendulum motion is SHM for small angles.

Period

Since:

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

the period is:

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

The period depends on pendulum length $l$ and gravitational field strength $g$. It does not depend on bob mass, and it is independent of amplitude only for small oscillations.

Key Trends

A longer pendulum has larger $T$ and oscillates more slowly. A stronger gravitational field gives smaller $T$ and faster oscillation.

A pendulum clock taken to a mountain where $g$ is slightly smaller will have a larger period and run slow.

Energy in Pendulum Motion

Ignoring air resistance, total mechanical energy is constant.

At highest points:

• speed is zero;
• kinetic energy is zero;
• gravitational potential energy is maximum.

At the lowest point:

• speed is maximum;
• kinetic energy is maximum;
• gravitational potential energy is minimum.

As the pendulum swings:

$$E_K\leftrightarrow E_P$$

Large-Angle Motion

If amplitude is large:

$$\sin \theta \ne \theta$$

Then motion is no longer exact SHM. The period becomes slightly larger and the motion is no longer perfectly sinusoidal.

Measuring g Experimentally

Rearrange:

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

Square both sides:

$$T^2=\frac{4{\pi }^2}{g}l$$

Plotting $T^2$ against $l$ gives a straight line with gradient:

$$\frac{4{\pi }^2}{g}$$

so:

$$g=\frac{4{\pi }^2}{\text{gradient}}$$

Common Mistakes

• Using the pendulum formula for large amplitudes.
• Thinking a heavier bob swings faster.
• Measuring length to the top or bottom of the bob instead of its centre of mass.
• Forgetting local $g$ affects period.
• Confusing arc displacement with vertical height.

Links

• Main topic: Oscillations and Simple Harmonic Motion
• Related concept: Simple Harmonic Motion
• Related concept: Damping and Resonance
• Related topic: Circular Motion
• Related topic: Gravitational Fields