16.4 The Simple Pendulum

Simple Pendulum Mechanics

A simple pendulum is a bob (treated as a point mass) suspended by a massless string from a fixed point. When you pull it to one side and release it, gravity acts as a restoring force that pulls it back toward equilibrium, causing it to swing back and forth. For small angles of displacement (less than about 15°), this restoring force is approximately proportional to the displacement, which is exactly what defines a harmonic oscillator.

Two things that don't affect the period might surprise you: the mass of the bob and the amplitude of the swing. As long as the angle stays small, the period stays constant regardless of how heavy the bob is or how far you pull it back.

The motion of the pendulum is described by this differential equation:

$\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0$

• $\theta$ is the angular displacement from equilibrium
• $L$ is the length of the pendulum (from the pivot to the center of the bob)
• $g$ is the acceleration due to gravity ($9.81 \text{ m/s}^2$ on Earth's surface)

This equation has the same form as the equation for a mass on a spring, which is why the pendulum qualifies as a simple harmonic oscillator under the small-angle approximation.

Calculation of Pendulum Period

The period ($T$) is the time for one complete back-and-forth oscillation. It depends only on the pendulum's length and the local gravitational acceleration:

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

To calculate the period:

1. Measure the length $L$ from the suspension point to the center of the bob.
2. Determine $g$ at the pendulum's location (use $9.81 \text{ m/s}^2$ unless told otherwise).
3. Plug $L$ and $g$ into the formula and solve.

Example: A pendulum with $L = 1$ m on Earth:

$T = 2\pi\sqrt{\frac{1}{9.81}} \approx 2.01 \text{ s}$

The angular frequency ($\omega$) is related to the period by:

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

This tells you how rapidly the pendulum cycles through its oscillation in radians per second.

Length vs. Period in Pendulums

The period is proportional to the square root of the length:

$T \propto \sqrt{L}$

This means the relationship is not linear. Doubling the length does not double the period. Instead:

• Double the length → period increases by a factor of $\sqrt{2} \approx 1.41$
  • A 1 m pendulum with a ~2.01 s period becomes a 2 m pendulum with a ~2.84 s period.
• Halve the length → period decreases by a factor of $\sqrt{2}$
  • A 1 m pendulum with a ~2.01 s period becomes a 0.5 m pendulum with a ~1.42 s period.

This square-root relationship is why grandfather clocks use adjustable pendulum lengths for precise timekeeping. Turning a small nut at the bottom of the bob raises or lowers the center of mass, fine-tuning the period.

Energy and Forces in a Pendulum

As the pendulum swings, energy converts continuously between two forms:

• At the highest points of the swing (maximum displacement), the bob has maximum gravitational potential energy and minimum kinetic energy. It momentarily stops before reversing direction.
• At the lowest point (equilibrium), the bob moves fastest, so it has maximum kinetic energy and minimum potential energy.

Total mechanical energy stays constant in an ideal pendulum, since gravity is a conservative force.

The tension in the string is not constant throughout the swing. It's greatest at the bottom, where it must support the bob's weight and provide the centripetal force for circular motion. At the top of the swing, the tension is at its minimum.

In real-world pendulums, damping from air resistance and friction at the pivot causes the amplitude to gradually decrease over time. The pendulum doesn't change its period noticeably as it damps down (another useful property for clocks), but it will eventually stop unless energy is added to the system.