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LearnAQA7408First teaching 2015Paper 1 · Paper 1Periodic motion

Paper 1 · Module

Periodic motion

Circular motion and simple harmonic motion (section 3.6.1)

5 subtopics≈ 175 min in total

Start the module: Kinematics of circular motion Open graphing calculator

Kinematics of circular motion
Angles in circular motion are measured in radians: $\theta=\dfrac sr$ , so a full turn is $2\pi$ rad. Angular speed $\omega$ is the rate of change of angle, $\omega=\dfrac{2\pi}T=2\pi f$ . A point at radius $r$ moves with linear speed $v=r\omega$ . Every point on a rigid rotating object has the same $\omega$ , but points further out move faster.
≈ 30 min
Simple harmonic motion
An oscillation is simple harmonic when the acceleration is proportional to the displacement from a fixed point and always directed towards that point: $a=-\omega^2x$ . The angular frequency is $\omega=2\pi f=\dfrac{2\pi}T$ . The solution is sinusoidal: $x=x_0\sin\omega t$ (starting from equilibrium). The period doesn't depend on the amplitude. A mass on a spring and a small-angle pendulum are both approximately simple harmonic.
≈ 35 min
Centripetal acceleration and force
An object moving in a circle accelerates towards the centre: $a=\dfrac{v^2}r=r\omega^2$ . This needs a resultant force towards the centre, $F=\dfrac{mv^2}r=mr\omega^2$ , called the centripetal force. It isn't a new kind of force: tension, gravity, friction or a normal reaction provides it. Being perpendicular to the velocity, it changes direction but does no work.
≈ 40 min
Velocity and energy in SHM
In SHM the speed is greatest at equilibrium, $v_0=\omega x_0$ , and zero at the extremes. At displacement $x$ , $v=\pm\omega\sqrt{x_0^2-x^2}$ . Energy continually swaps between kinetic and potential, but the total stays constant (with no damping): $E=\tfrac12m\omega^2x_0^2$ . Kinetic energy is $\tfrac12m\omega^2(x_0^2-x^2)$ and potential energy is $\tfrac12m\omega^2x^2$ .
≈ 35 min
Damping and resonance
A resistive force (friction, air or fluid resistance) removes energy from an oscillator, so its amplitude decays: damping. Light damping gives slowly decaying oscillations; critical damping returns the system to equilibrium in the shortest time without oscillating; heavy damping returns it slowly without oscillating. When a periodic driving force matches the natural frequency, the amplitude becomes a maximum: resonance. More damping lowers and broadens the resonance peak.
≈ 35 min