Energy Exchange in SHM

In simple harmonic motion (SHM), energy is constantly swapping between two forms: kinetic energy (the energy of movement) and elastic potential energy (the energy stored by displacement from equilibrium). Think of a playground swing: at the bottom it moves fastest and has maximum kinetic energy, while at the highest points it momentarily stops and all its energy is potential. This continuous exchange is the heartbeat of every oscillating system, from a mass on a spring to the vibrating atoms in a crystal. The remarkable feature of ideal SHM is that the total mechanical energy — the sum of kinetic and potential energy — remains perfectly constant throughout the motion, provided no friction or air resistance acts. This conservation of mechanical energy means that whatever kinetic energy is lost as the object slows down is exactly gained as potential energy, and vice versa. The total energy depends only on the amplitude A and the spring constant k (or equivalently the mass m and angular frequency ω), not on the instantaneous position or velocity. Understanding this energy exchange is crucial for analysing real oscillating systems such as clock pendulums, guitar strings, and suspension bridges. Even when damping is present in real life, the underlying energy interchange described here forms the baseline model from which more complex behaviour is understood.

Key takeaways

Setting Up the Energy Expressions
Writing the Conservation Law
Finding Velocity at Any Position
Energy as a Function of Time
Graphical Interpretation

Setting Up the Energy Expressions

For a mass m attached to a spring of spring constant k, displaced by x from equilibrium and moving with velocity v, the elastic potential energy is U = ½kx² and the kinetic energy is K = ½mv². These are the only two forms of mechanical energy involved in ideal SHM.

Writing the Conservation Law

Because no energy is lost in ideal SHM, the total energy E = K + U = ½mv² + ½kx² is constant at all times. At the amplitude position x = A (or x = −A), velocity is zero so E = ½kA². At the equilibrium x = 0, potential energy is zero so E = ½mv²_max. Setting these equal gives v_max = A√(k/m) = Aω, where ω = √(k/m) is the angular frequency.

Finding Velocity at Any Position

Since E = ½kA² at all times, we can write ½mv² + ½kx² = ½kA², which rearranges to v = ω√(A² − x²). This powerful result lets you calculate the speed at any displacement x without needing to track time explicitly. Notice that speed is maximum at x = 0 and zero at x = ±A, confirming our physical intuition about swings and springs.

Energy as a Function of Time

Using the SHM position function x = A cos(ωt), the potential energy varies as U(t) = ½kA²cos²(ωt) and the kinetic energy as K(t) = ½kA²sin²(ωt). Both oscillate between 0 and ½kA² but are exactly out of phase by 90° (a quarter period). Their sum always equals ½kA², confirming constant total energy. The energy oscillates at twice the frequency of the displacement itself.

Graphical Interpretation

If you plot U and K against displacement x, U is a parabola opening upward (½kx²) and K is an inverted parabola, also described by ½k(A²−x²). They are mirror images that add to give the constant horizontal line E = ½kA². This parabolic potential energy well is a universal feature of any system undergoing SHM — the shape of the potential energy curve near any stable equilibrium is approximately parabolic, which is why SHM is so common in nature.

The Key Equation

E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2

E — Total mechanical energy (J (joules)): The constant total energy of the oscillating system, determined by its amplitude and spring constant.
m — Mass (kg): The mass of the oscillating object attached to the spring.
v — Instantaneous velocity (m s⁻¹): The speed of the mass at displacement x; zero at x = ±A and maximum at x = 0.
k — Spring constant (N m⁻¹): The stiffness of the spring; a larger k means more restoring force per unit displacement.
x — Displacement (m): The instantaneous distance of the mass from the equilibrium position.
A — Amplitude (m): The maximum displacement from equilibrium; sets the total energy of the system.

Worked Example

Problem. A 0.200 kg mass oscillates on a spring of spring constant k = 50.0 N m⁻¹ with an amplitude of 0.080 m. (a) Calculate the total mechanical energy of the system. (b) Find the speed of the mass when it is at x = 0.060 m from equilibrium. (c) Find the maximum speed.

E = 0.160\text{ J},\quad v_{x=0.06} \approx 0.837\text{ m s}^{-1},\quad v_{\max} = A\omega \approx 1.26\text{ m s}^{-1}

Answer. Total energy = 0.160 J; speed at x = 0.060 m is approximately 0.837 m s⁻¹; maximum speed ≈ 1.26 m s⁻¹.

Figure

Diagram

Origins · 1 minRobert Hooke and how this was discovered

Robert Hooke

1635–1703 · English

Robert Hooke formulated the law bearing his name — that the restoring force in a spring is proportional to its displacement (F = −kx) — which is the foundational relationship underlying all of SHM energy analysis. Published in anagram form in 1676 and openly in 1678, Hooke's Law defines the parabolic potential energy well U = ½kx² whose continuous exchange with kinetic energy characterises every simple harmonic oscillator. Hooke originally published his spring law as the Latin anagram 'ceiiinosssttuv' to protect his priority — it unscrambles to 'ut tensio sic vis', meaning 'as the extension, so the force'.

Introduction to SHM — SHM energy exchange builds directly on understanding displacement, velocity, and acceleration in simple harmonic motion.
SHM Equations of Motion — The position function x = A cos(ωt) and velocity function v = −Aω sin(ωt) are used to derive the time-varying expressions for kinetic and potential energy in SHM.
Damped Oscillations — Damping causes the total mechanical energy to decrease over time, showing what happens when the ideal energy conservation of SHM is broken by dissipative forces.
Wave Energy and Intensity — The energy carried by a wave depends on the square of its amplitude, a direct consequence of the SHM energy relationship E = ½kA².