The Simple Pendulum, Damping and Resonance: Quick Revision

A brief note on The Simple Pendulum, Damping and Resonance. Read it before a test. Try each question at the end before you open its answer.

The simple pendulum

• Pull back $mg\sin\theta \approx mg\theta$ for a small angle, in radians.
• $T = 2\pi\sqrt{L/g}$. No mass in it, and no small amplitude.
• Seconds pendulum: $T = 2\ \text{s}$, $L \approx 1\ \text{m}$ on Earth.
• Changed gravity: use $g_{\text{eff}}$. Lift speeding up upwards, $g + a$. Downwards, $g - a$. Free fall, $0$, so no swing.
• Wide swings take a little longer than the formula says.

Other oscillators

• Rule: pull back in proportion to displacement gives SHM.
• U-tube: $T = 2\pi\sqrt{L/2g}$, with $L$ the whole column. Density does not matter.
• Floating body: $T = 2\pi\sqrt{h/g}$, with $h$ the depth under the surface.
• Physical pendulum: $T = 2\pi\sqrt{I/mgd}$. A rod about one end: $T = 2\pi\sqrt{2L/3g}$.

Damping and resonance

• Damping force: $-bv$.
• Amplitude: $A_0e^{-bt/2m}$, the same factor in each equal time.
• Energy: $\propto e^{-bt/m}$. Half the amplitude means a quarter of the energy.
• Time to halve the amplitude: $\dfrac{2m\ln 2}{b}$.
• Forced oscillation: steady motion at the driving frequency.
• Resonance: largest amplitude when the driving frequency equals the natural frequency. Less damping, a sharper peak.

A worked example

A swing on chains $1.8\ \text{m}$ long is taken to the Moon, where $g$ is a sixth of its value on Earth.

1. On Earth. $T = 2\pi\sqrt{\dfrac{1.8\ \text{m}}{9.8\ \text{m/s}^2}} = 2.69\ \text{s}$.
2. On the Moon. $g$ is six times smaller, so $T$ is $\sqrt6$ times larger: $6.60\ \text{s}$.
3. To keep it going. Push once in each $6.60\ \text{s}$. That is resonance.

Practice with new numbers

A program to try

The program finds the length of a seconds pendulum in two places.

Mistakes to avoid

• Putting the mass into $T$. It cancels.
• Using $g$ for the U-tube. It is $2g$.
• A straight-line fall of amplitude. It falls by the same factor in equal times.
• Saying a forced body swings at its own frequency. It swings at the driving frequency.

Check yourself

1. How long is a seconds pendulum on Earth?

2. What is the period of the liquid in a U-tube with a column $L$ long?

3. The amplitude of an oscillator falls to a third. What fraction of the energy is left?

4. Why do soldiers break step on a bridge?