Energy in oscillations and dampingEdexcel International A Level Physics: Revision notes

An oscillator exchanges energy between two stores. For a mass on a spring the stores are kinetic energy and elastic potential energy; for a pendulum they are kinetic energy and gravitational potential energy. In an undamped system no energy is transferred to the surroundings, so the total energy is constant and the amplitude does not change.

The kinetic energy is maximum at the equilibrium position, where the speed is greatest and the potential energy is zero. At the amplitude (the maximum displacement) the oscillator is momentarily at rest, so the kinetic energy is zero and the potential energy is maximum. Conservation of energy therefore gives total energy = maximum kinetic energy = maximum potential energy.

For SHM with angular frequency ω = 2π/T = 2πf, the speed at displacement x is v = ±ω√(A² − x²). Hence:

• kinetic energy: $E_k = \frac{1}{2}m\omega^2(A^2 - x^2)$
• potential energy: $E_p = \frac{1}{2}m\omega^2 x^2$
• total energy: $E = \frac{1}{2}m\omega^2 A^2$

For a spring, $\omega^2 = k/m$, so $E = \frac{1}{2}kA^2$ and $E_p = \frac{1}{2}kx^2$. The total energy is proportional to (amplitude)²: doubling the amplitude quadruples the energy.

Worked example. A 0.40 kg trolley on a spring with k = 64 N m⁻¹ has amplitude 0.050 m. Total energy = ½ × 64 × 0.050² = 0.080 J. Maximum speed: ½mv² = 0.080 J, so v = √(2 × 0.080 / 0.40) = 0.63 m s⁻¹. At x = 0.030 m, $E_p = \frac{1}{2} \times 64 \times 0.030^2 = 0.029$ J, so $E_k = 0.080 - 0.029 = 0.051$ J.

In a damped oscillator a resistive force (air resistance, viscous drag, friction) acts against the motion. The oscillator does work against it, so mechanical energy is transferred to thermal energy of the surroundings. The total energy of the oscillation decreases each cycle and, because E ∝ A², the amplitude decreases with time.

For light damping the amplitude falls gradually, roughly by the same fraction each cycle, while the period is almost unchanged. Heavier damping removes the energy faster. Energy is never destroyed: the total energy of oscillator plus surroundings is conserved.

Light (under) damping: the system oscillates with gradually decreasing amplitude. Critical damping: the system returns to equilibrium in the shortest time without oscillating, as in a car suspension or a door closer. Heavy (over) damping: the system returns to equilibrium very slowly without oscillating.

A pendulum in air is lightly damped; a mass in thick oil may be heavily damped. Engineers choose the degree of damping to suit the purpose of the system.

A ductile material such as steel can be stretched far beyond its elastic limit. Past this point it deforms plastically: the deformation is permanent and the energy used to produce it is not recovered when the force is removed. On a force–extension (or stress–strain) graph the loading and unloading lines enclose a loop, and the area of the loop is the energy transferred to thermal energy in the material during that cycle.

In buildings and bridges, ductile members that yield during an earthquake dissipate energy and reduce the amplitude of the swaying, which helps prevent collapse. Together with viscous dampers they act as damping, reducing the amplitude of oscillation.

Three habits earn marks. First, name the stores involved and say that energy is exchanged between them. Second, in a damped system, state that work is done against a resistive force, which transfers mechanical energy to thermal energy, so the amplitude decreases. Third, compare energies using E ∝ A²: if the amplitude falls from 6.0 cm to 3.0 cm the energy falls to one quarter, so 75% has been transferred away.

For calculations, find ω from 2π/T or 2πf first, then substitute into E = ½mω²A².