Energy, damping and resonanceEdexcel A-Level Physics: Revision notes

In an undamped oscillator no energy is lost, so the total energy is constant. It is continuously transferred between kinetic energy and potential energy (elastic potential for a mass on a spring, gravitational potential for a pendulum).

• At maximum displacement (the amplitude A): velocity is zero, so all the energy is potential.
• At the equilibrium position: potential energy is a minimum and kinetic energy is a maximum.
• $E_k = \frac{1}{2} m \omega^2 (A^2 - x^2)$ and $E_p = \frac{1}{2} m \omega^2 x^2$, so $E_{total} = \frac{1}{2} m \omega^2 A^2$.
• The total is proportional to A²: doubling the amplitude gives four times the energy.

Worked example: a 0.50 kg trolley oscillates with $A = 0.060$ m and $T = 1.2$ s. Then $\omega = 2\pi/T = 5.24$ rad s⁻¹ and $E_{k,max} = \frac{1}{2} \times 0.50 \times 5.24^2 \times 0.060^2 = 0.025$ J.

A free oscillation occurs when a system is displaced and released, and then oscillates with no external force acting. It oscillates at its natural frequency $f_0$, which depends only on the system (for a mass on a spring $f_0 = \frac{1}{2\pi}\sqrt{k/m}$).

A forced oscillation occurs when a periodic driving force acts on the system. After the initial transient the system oscillates at the driving frequency, not at its natural frequency.

• The driver does work on the system and replaces energy lost to the surroundings.
• The amplitude of the forced oscillation depends on how close the driving frequency is to the natural frequency.

Resonance occurs when the driving frequency equals the natural frequency of the system. The amplitude of the forced oscillation is then a maximum, because energy is transferred from the driver to the system at the greatest possible rate.

On a graph of amplitude against driving frequency the curve rises to a peak at the natural frequency and falls on either side. The driving force is in phase with the velocity of the system at resonance, so the driver does positive work during the whole cycle.

Examples: a child on a swing pushed at the right moment, a washing machine drum shaking at one spin speed, a bridge or building driven by wind or footsteps.

Damping is the loss of energy from an oscillating system through resistive forces (such as air resistance, friction or viscous drag), which do work against the motion. The lost energy is dissipated as thermal energy in the surroundings.

• A damped free oscillation has an amplitude that decreases with each cycle.
• Light damping: many oscillations before the motion dies away.
• Critical damping: the system returns to equilibrium in the shortest time with no oscillation (for example car shock absorbers, door closers).
• Heavy (over) damping: the system returns to equilibrium slowly without oscillating.

For a forced oscillation, damping lowers the peak at resonance, makes the resonance curve broader, and moves the peak slightly towards a lower frequency. Plastic deformation of a ductile material in a structure also dissipates energy, so it too reduces the amplitude of oscillation (but leaves the structure permanently distorted).

A mass hung on a spring oscillates with $T = 2\pi \sqrt{m/k}$, so $T^2 = \frac{4\pi^2}{k} m$. For a real spring, the spring's own mass adds a positive intercept: $T^2 = \frac{4\pi^2}{k} m + c$.

1. Hang known masses from the spring in turn and displace each through a small amplitude.
2. Time 10 or 20 oscillations from the equilibrium position (using a fiducial marker), and divide to find T. Repeat and average.
3. Plot $T^2$ against $m$ and draw a line of best fit. The gradient is $4\pi^2 / k$.
4. Hang the unknown mass, measure its T, and read its mass from the graph (or calculate $m = (T^2 - c) / \text{gradient}$).

Timing many oscillations reduces the percentage uncertainty from reaction time, and a small amplitude keeps damping small and the motion simple harmonic.