Energy in SHM: Kinetic, Potential and Total Energy

An ideal pendulum never runs down; it trades kinetic and potential energy back and forth. At the top of each swing it is momentarily still but high up; at the bottom it is low but racing. The total energy stays constant while kinetic and potential energy take turns.

This energy exchange explains why the period ignores amplitude, how to find speed at any point, and why real oscillators die out as friction steals energy each cycle.

Key Concepts

• Kinetic energy in SHM: KE = (1/2) m v^2 = (1/2) m omega^2 (A^2 - x^2). It is maximum at the mean position (x = 0) and zero at the extremes (x = plus or minus A).
• Potential energy in SHM: PE = (1/2) k x^2 = (1/2) m omega^2 x^2, taking the mean position as zero potential. It is maximum at the extremes and zero at the centre.
• Total mechanical energy: E = KE + PE = (1/2) k A^2 = (1/2) m omega^2 A^2, constant at every point of the motion when there is no friction.
• Damping: in real systems friction converts a little mechanical energy into heat each cycle, so the amplitude slowly shrinks while the frequency stays nearly the same.

Formulas

• Kinetic energy: KE = (1/2) m omega^2 (A^2 - x^2). Units: joules.
• Potential energy: PE = (1/2) m omega^2 x^2, with k = m omega^2.
• Total energy: E = (1/2) m omega^2 A^2 = (1/2) k A^2. It depends on the square of the amplitude, so doubling the amplitude quadruples the energy.
• Speed at displacement x: v = omega sqrt(A^2 - x^2), found by equating KE to E - PE.
• Assumptions and limits: energy is conserved only without damping. The formulas assume the linear restoring force of true SHM.

Worked Example

A 0.2 kg mass oscillates on a spring with amplitude 10 cm and angular frequency 5 rad/s. Find the total energy, and the kinetic and potential energies when the displacement is 5 cm.

1. Total energy E = (1/2) m omega^2 A^2 = 0.5 x 0.2 x 25 x 0.01 = 0.025 J.
2. Force constant k = m omega^2 = 0.2 x 25 = 5 N/m.
3. At x = 0.05 m: PE = (1/2) k x^2 = 0.5 x 5 x 0.0025 = 0.00625 J.
4. KE = E - PE = 0.025 - 0.00625 = 0.01875 J, and KE + PE = 0.025 J, the constant total. At the mean position v_max = omega A = 0.5 m/s.

Common Misconceptions

• Energy is not maximum at the extremes. The total energy is the same everywhere; only its form changes. At the extremes it is all potential, at the centre all kinetic.
• Doubling the amplitude does not double the energy. Energy goes as amplitude squared, so doubling A quadruples E. The extra distance means the spring stores four times the work.

Real-World Applications

• Engineers size shock absorbers by the energy they must soak up each bounce, which scales with amplitude squared.
• Quartz watches keep time because the crystal's steady energy exchange gives a rock-stable frequency.
• Playground swing designers know a higher swing stores far more energy, which is why safety limits target amplitude.
• Seismic dampers in tall buildings absorb earthquake energy by converting the building's vibrational energy into heat.

In SHM, kinetic energy peaks at the centre as (1/2) m omega^2 (A^2 - x^2) while potential energy peaks at the ends as (1/2) m omega^2 x^2, and their sum stays locked at (1/2) m omega^2 A^2. Energy simply shuttles between motion and the spring, which is why frictionless oscillation runs forever.