Simple Harmonic Motion: Quick Revision

A brief note on Simple Harmonic Motion. Read it before a test. Try each question at the end before you open its answer.

Periodic motion

• A periodic motion repeats after a fixed time, the period $T$, in seconds.
• An oscillation is a periodic motion to and fro about a mean position.
• Frequency: $f = \dfrac{1}{T}$, in hertz.

The equation of SHM

• $x = A\cos(\omega t + \phi)$.
• $A$ is the amplitude, $\omega t + \phi$ the phase, $\phi$ the phase constant.
• Angular frequency: $\omega = \dfrac{2\pi}{T} = 2\pi f$, in rad/s.
• SHM is the foot, on a diameter, of a point moving round a circle of radius $A$ at a steady speed.

Velocity and acceleration

• $v = -\omega A\sin(\omega t + \phi)$. At the position $x$, $|v| = \omega\sqrt{A^2 - x^2}$.
• $a = -\omega^2 x$: always towards the mean position.
• At the mean position: $|v| = \omega A$ is largest and $a = 0$.
• At the ends: $v = 0$ and $|a| = \omega^2 A$ is largest.
• $v$ leads $x$ by $\pi/2$. $a$ leads $x$ by $\pi$.

A worked example

A body moves in SHM with an amplitude of $3\ \text{cm}$ at $4\ \text{Hz}$. Find its largest speed and largest acceleration.

1. Angular frequency. $\omega = 2\pi \times 4\ \text{Hz} = 25.1\ \text{rad/s}$.
2. Largest speed. $v_{max} = 25.1\ \text{rad/s} \times 0.03\ \text{m} = 0.754\ \text{m/s}$.
3. Largest acceleration. $a_{max} = (25.1\ \text{rad/s})^2 \times 0.03\ \text{m} = 18.9\ \text{m/s}^2$.

Practice with new numbers

A program to try

The program works through the example above, and adds the speed at half the amplitude.

Mistakes to avoid

• Mixing up $f$ and $\omega$. $\omega = 2\pi f$.
• Degrees in $\cos\omega t$. The phase is in radians.
• Largest acceleration at the centre. It is largest at the ends.
• Taking the whole width of the path as $A$. The width is $2A$.

Check yourself

1. A motion repeats $50$ times in $10\ \text{s}$. Find its period and frequency.

2. In $x = 6\cos(\pi t)$ cm, what is the period?

3. Where is a body in SHM when its acceleration is largest?

4. A body in SHM has $A = 5\ \text{cm}$ and $\omega = 4\ \text{rad/s}$. What is its speed at $x = 3\ \text{cm}$?