KAHIBARO
Table of Contents
Basic idea
A simple pendulum is one of the most important examples of periodic motion. It consists of a small mass, called the bob, attached to a light string of length $L$, with the other end fixed. The bob swings back and forth under the influence of gravity.
The word simple means that we make an idealized model. We assume that the string has no mass, the bob is treated like a point particle, the motion happens in a single vertical plane, and air resistance is neglected. We also assume that the pivot is frictionless.
When the bob is pulled slightly away from its lowest point and released, it oscillates about the equilibrium position. The restoring effect is provided by gravity.
Geometry of the motion
The position of the bob is usually described by the angle $\theta$ that the string makes with the vertical. The lowest point corresponds to $\theta = 0$.
As the pendulum swings, the bob moves along a circular arc of radius $L$. The distance traveled along the arc from equilibrium is
$$
s = L\theta
$$
when $\theta$ is measured in radians.
This means that angular displacement and arc displacement are directly related.
Restoring force
The weight of the bob acts vertically downward with magnitude $mg$. Part of this force is balanced by the string tension. The part that causes the pendulum to swing is the tangential component of gravity.
That tangential component is
$$
F_t = -mg\sin\theta
$$
The negative sign shows that the force always points toward the equilibrium position. If $\theta$ is positive, the force is negative. If $\theta$ is negative, the force is positive.
For a simple pendulum, the restoring force along the arc is
$$
F_t = -mg\sin\theta
$$
This force is responsible for the oscillation.
Equation of motion
Using Newton's second law along the arc,
$$
m a_t = -mg\sin\theta
$$
The tangential acceleration is related to angular acceleration by
$$
a_t = L\frac{d^2\theta}{dt^2}
$$
So the exact equation of motion becomes
$$
mL\frac{d^2\theta}{dt^2} = -mg\sin\theta
$$
or
$$
\frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0
$$
This equation describes the pendulum exactly for the ideal model.
Small-angle approximation
The exact pendulum equation is not the same as the standard simple harmonic motion equation because of the $\sin\theta$ term. However, when the angle is small and measured in radians, we can use the approximation
$$
\sin\theta \approx \theta
$$
Then the equation becomes
$$
\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0
$$
This is exactly the equation of simple harmonic motion.
A simple pendulum behaves like simple harmonic motion only for small angles, where
$$
\sin\theta \approx \theta
$$
Then
$$
\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0
$$
This is why the simple pendulum is such an important physical example of SHM.
Angular frequency and period
Comparing the pendulum equation in the small-angle limit with the SHM form
$$
\frac{d^2x}{dt^2} + \omega^2 x = 0
$$
we identify
$$
\omega = \sqrt{\frac{g}{L}}
$$
From this, the period is
$$
T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{L}{g}}
$$
and the frequency is
$$
f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{g}{L}}
$$
These formulas are valid for small oscillations.
For a simple pendulum at small angles,
$$
\omega = \sqrt{\frac{g}{L}}
$$
$$
T = 2\pi\sqrt{\frac{L}{g}}
$$
$$
f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}
$$
The period does not depend on the mass of the bob.
What affects the period
The formula
$$
T = 2\pi\sqrt{\frac{L}{g}}
$$
shows two important facts. A longer pendulum has a larger period, so it swings more slowly. A stronger gravitational field gives a smaller period, so it swings more quickly.
The mass of the bob does not appear in the formula. In the ideal model, heavy and light bobs with the same length have the same period.
The amplitude also does not appear in the small-angle formula. This means that for small oscillations, the period is approximately independent of amplitude. For larger angles, this is no longer exactly true.
Comparison table
| Quantity | Symbol | Formula for small angles |
|---|---|---|
| Arc displacement | $s$ | $s = L\theta$ |
| Restoring force | $F_t$ | $-mg\sin\theta$ |
| Angular frequency | $\omega$ | $\sqrt{g/L}$ |
| Period | $T$ | $2\pi\sqrt{L/g}$ |
| Frequency | $f$ | $\frac{1}{2\pi}\sqrt{g/L}$ |
Why the simple pendulum is useful
Because its period depends mainly on length and gravity, the simple pendulum can be used to measure time and to estimate the local value of $g$. Rearranging the period formula gives
$$
g = \frac{4\pi^2 L}{T^2}
$$
So if the length and period are measured, the gravitational acceleration can be found.
A pendulum can be used to determine gravitational acceleration:
$$
g = \frac{4\pi^2 L}{T^2}
$$
This relation is valid when the oscillations are small.
Limits of the model
The simple pendulum is an idealization. Real pendulums are affected by air resistance, friction at the pivot, and the finite size of the bob. Also, if the angle becomes large, the approximation $\sin\theta \approx \theta$ is no longer accurate, and the motion is not exactly simple harmonic.
So the formula
$$
T = 2\pi\sqrt{\frac{L}{g}}
$$
is extremely useful, but it should be applied under the right conditions.
Visualizing the restoring effect
Final picture
A simple pendulum is a mass on a light string that swings under gravity. Its exact motion is governed by
$$
\frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0
$$
For small angles, this becomes SHM, and the period is
$$
T = 2\pi\sqrt{\frac{L}{g}}
$$
This result is one of the classic formulas in physics because it is simple, useful, and experimentally important.