Periodic motion

What periodic motion covers

Motion that repeats: going round a circle at steady speed, and oscillating about an equilibrium position. The unit supplies the angular quantities that gravitational fields, magnetic fields and the engineering physics option all use later. AQA's required practical on the simple pendulum is examined from here.

The main ideas

• Why constant-speed circular motion is accelerated motion, and what a centripetal force is.
• Radians, and angular speed related to speed and to frequency.
• Centripetal acceleration and force, and naming the real force that supplies it each time.
• The condition for simple harmonic motion: acceleration proportional to displacement and directed back towards the centre.
• The solutions for displacement and velocity, where the maxima occur, and the three graphs linked by their gradients.
• Mass-spring and pendulum periods, what each depends on and what neither depends on.
• The exchange of kinetic and potential energy, damping of three kinds, and forced vibration with resonance.

The equations it turns on

ω = vr = 2πf$\omega = \frac{v}{r} = 2\pi f$ω = v/r = 2πf

angular speed, in radians per second

a = v²r = ω²r     F = mv²r = mω²r$a = \frac{v^2}{r} = \omega^{2}r \qquad F = \frac{mv^2}{r} = m\omega^{2}r$a = v^2/r = ω^2r F = mv^2/r = mω^2r

centripetal acceleration and the force producing it

a = -ω²x$a = -\omega^{2}x$a = -ω^2x

the defining equation of simple harmonic motion

x = Acos (ωt)     v = ±ω√A² - x²$x = A\cos(\omega t) \qquad v = \pm\omega\sqrt{A^2 - x^2}$x = A cos(ωt) v = ±ω√(A^2 - x^2)

position and speed at any displacement

T = 2π√m/k$T = 2\pi\sqrt{m/k}$T = 2π√(m/k)

the period of a mass on a spring

T = 2π√l/g$T = 2\pi\sqrt{l/g}$T = 2π√(l/g)

the period of a simple pendulum, small angles only

Where it usually goes wrong

• There is no outward force on an object moving in a circle. The force is inward and always has a name: friction, tension, gravity or a contact force. Remove it and the object travels in a straight line, not outwards.
• The minus sign in a = -omega^2 x carries the physics. Displacement is measured from the centre and the acceleration points back to it, so their signs are always opposite.
• The period does not depend on amplitude, the pendulum period does not depend on the mass of the bob, and the mass-spring period does not depend on g.
• Heavier damping lowers the peak of a resonance curve and broadens it, so the response spreads across a wider range of driving frequencies.

Where to start

Circular motion first, because it introduces radians and angular speed. Simple harmonic motion next, then pendulums and springs, which put numbers on it. Forced vibrations and resonance last.