The simple pendulum

Why does a pendulum keep time?

Galileo watched a lamp swing in a church. Some swings were wide and some were narrow. He timed them with his pulse, and they all took the same time. That is why a pendulum can run a clock.

In this lesson we find the period of a pendulum. We also find what changes it and what does not.

The simple pendulum

A simple pendulum is a small heavy bob on a light string that does not stretch. The string has length $L$, from the support to the centre of the bob. Pull the bob aside by an angle $\theta$ and let go.

Two forces act on the bob: the tension $T$ along the string, and its weight $mg$. Split the weight into two parts:

• $mg\cos\theta$ along the string. The tension balances it, and also gives the pull towards the centre that a curved path needs.
• $mg\sin\theta$ along the path, towards the middle. This part pulls the bob back.

The pull back gives a torque about the support:

The minus sign says the torque turns the bob back towards $\theta = 0$.

The small angle step

For a small angle, $\sin\theta \approx \theta$ in radians. At $10^\circ$ the two differ by less than $1\ \%$. So

The bob has moment of inertia $I = mL^2$ about the support. Newton's second law for turning is $\tau = I\alpha$, so

The angular acceleration is in proportion to the angle, and opposite to it. That is simple harmonic motion, with $\omega^2 = g/L$. So the period is

Notice what is missing. The mass cancelled out. The amplitude is not there either, as long as it is small.

A worked example. A playground swing hangs on chains $2.5\ \text{m}$ long. Treat it as a simple pendulum.

The seconds pendulum

A seconds pendulum takes one second for each swing from one side to the other. So its period is $2\ \text{s}$. We want $L$, so square the formula and solve:

That is close to one metre. Old clocks often had a pendulum about this long.

When g changes

The period depends on the pull of gravity. On the Moon, $g$ is about one sixth of its value on Earth. So the same pendulum swings $\sqrt6$ times slower there, about two and a half times.

In a lift. Suppose a lift speeds up while going up, with acceleration $a$. The bob feels a larger pull, as if gravity were $g + a$. If the lift speeds up going down, it feels $g - a$. So we write

where $g_{\text{eff}}$ is the gravity the bob feels. Take a pendulum $1.0\ \text{m}$ long. At rest, $T = 2.01\ \text{s}$. In a lift speeding up upwards at $1.8\ \text{m/s}^2$, $g_{\text{eff}} = 11.6\ \text{m/s}^2$ and $T = 1.84\ \text{s}$.

What if the lift falls freely? Then $g_{\text{eff}} = 0$. There is no pull back, and the pendulum does not swing at all.

In a car. A car that speeds up along a level road, or goes round a curve, also changes $g_{\text{eff}}$. The bob then hangs at an angle. It feels gravity and the sideways effect together, so $g_{\text{eff}} = \sqrt{g^2 + a^2}$. The period gets a little shorter.

Make the length four times as large. The period doubles. Now slide the gravity down to the Moon's value. The bob swings much more slowly. Then try a lift that speeds up upwards.

Wide swings

For a large angle, $\sin\theta$ is less than $\theta$. The pull back is a little weaker than the simple formula says, so the swing takes longer. At $20^\circ$ the period is less than $1\ \%$ longer. At $60^\circ$ it is about $7\ \%$ longer. For a clock we keep the swing small.

Practise with new numbers

Try it in Python

The program finds the period for three lengths, in three places.

Common mistakes

• Using the length of the string alone. $L$ goes to the centre of the bob.
• Thinking a heavier bob swings faster. The mass cancels. Only $L$ and $g$ matter.
• Using the formula for wide swings. $T = 2\pi\sqrt{L/g}$ is for small angles only.
• Forgetting the lift or the car. Use the gravity the bob feels, $g_{\text{eff}}$, not $g$.

Remember these

• Pull back: $mg\sin\theta \approx mg\theta$ for small angles.
• $T = 2\pi\sqrt{L/g}$, the same for any mass and any small amplitude.
• A seconds pendulum has $T = 2\ \text{s}$ and $L \approx 1\ \text{m}$ on Earth.
• In a lift: $g_{\text{eff}} = g + a$ going up faster, $g - a$ going down faster, $0$ in free fall.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A pendulum is made nine times as long. What happens to its period?

2. Find the period of a pendulum $0.25\ \text{m}$ long on Earth.

3. Why does the mass of the bob not change the period?

4. A lift moves down at a steady speed. Does a pendulum inside it swing faster, slower or the same?