
## Periodic motion

Periodic motion builds on the concepts that you learnt in the Year 12 mechanics module, but applies them to objects whose motion is either circular or periodic (e.g. pendulums). This is quite a difficult module, as you will need a good understanding of Newton’s III law and strong maths skills. You will study circular motion, simple harmonic motion and how to apply it to both pendulums and springs, resonance and damping. You will need to understand this concepts both mathematically and graphically. This module will be tested on Paper 1 of the A level exam.

The practical skills that you will develop in this module will focus on how to reduce uncertainty through careful design. You will be expected to measure oscillations that may have a short period so will have to develop strategies to reduce the uncertainty in these measurements. This module contains one CAP, Simple harmonic motion, for which you will be expected to plan the method.

### What you need to know

Below you can read exactly what AQA want you to know for this module. You can also find the relevant section from the specification on each page of this site. You should be aware of both what you need to know, and (just as importantly) what you DO NOT need to know. It is also important to remember that you need to be able to apply these statements to a wide range of different contexts, so you must practise this by attempting lots of different questions and reading around the subject.

3.6.1.1 Circular motion

Motion in a circular path at constant speed implies there is an acceleration and requires a centripetal force.

Magnitude of angular speed,

$$\omega=\frac{v}{r}=2\pi f$$

Direction of angular velocity will not be considered.

Centripetal acceleration,

$$a=\frac{v^{2}}{r}=\omega^{2}r$$

The derivation of the centripetal acceleration formula will not be examined.

Centripetal force,

$$F=\frac{mv^{2}}{r}=m\omega^{2}r$$

3.6.1.2 Simple harmonic motion

Analysis of characteristics of simple harmonic motion (SHM).

Condition for SHM: $a\propto -x$

Defining equation: $a=-\omega^{2}x$

$x=A\cos\omega t$ and $v=\pm \omega \sqrt{\left (A ^{2}-x^{2} \right )}$

Graphical representations linking the variations of x, v and a with time.

Appreciation that the v - t graph is derived from the gradient of the x - t graph and that the a - t graph is derived from the gradient of the v - t graph.

Maximum speed $=\omega A$

Maximum acceleration $=\omega^{2}A$

3.6.1.3 Simple harmonic systems

Study of mass-spring system:

$$T=2\pi \sqrt{\frac{m}{k}}$$

Study of simple pendulum: $$T=2\pi \sqrt{\frac{l}{g}}$$

Questions may involve other harmonic oscillators (eg liquid in U-tube) but full information will be provided in questions where necessary.

Variation of Ek, Ep, and total energy with both displacement and time.

Effects of damping on oscillations.

3.6.1.4 Forced vibrations and resonance

Qualitative treatment of free and forced vibrations.

Resonance and the effects of damping on the sharpness of resonance.

Examples of these effects in mechanical systems and situations involving stationary waves.