Energy in Simple Harmonic Motion

When you watch a mass attached to a spring bounce back and forth, it can almost seem like the motion continues on its own. The mass speeds up, slows down, stops, and reverses direction without anyone pushing it again. The reason this happens is energy. In Simple Harmonic Motion (SHM), energy is never destroyed in an ideal system. Instead, it continuously shifts back and forth between potential energy and kinetic energy in a perfectly predictable pattern.

In ideal SHM, meaning no friction or air resistance, the total mechanical energy of the system remains constant. Energy does not disappear or get created; it simply changes form. At every point in the motion, the system has some combination of potential energy (stored energy) and kinetic energy (energy of motion), but the total amount always stays the same.

The Big Idea: Energy Is Conserved

In ideal SHM (no friction, no air resistance): The total mechanical energy of the system stays constant. Energy does not disappear, nor does it get created; it simply changes form. In SHM, energy continuously shifts between Potential Energy (stored energy) and Kinetic Energy (energy of motion). One key detail is that the system’s total energy remains the same at every point in the motion.

Energy in a Mass–Spring System

At the maximum displacement, also called the amplitude, the mass is momentarily at rest before changing direction. Because its velocity is zero at that instant, its kinetic energy is zero. However, the spring is either stretched or compressed the most at this point, meaning the elastic potential energy stored in the spring is at its maximum. All of the system’s energy is stored in the spring. This means that through the formula Uk = 1/2(m)(v)^2, since velocity is 0, the kinetic energy is zero. On the other hand, through the formula Us = 1/2(k)(A)^2, since the block is at a maximum displacement of A, it has maximum potential energy, which is equal to the system's total energy.

As the mass moves toward equilibrium, the spring begins to return to its natural length. The stored elastic potential energy decreases while the kinetic energy increases. By the time the mass passes through equilibrium, the spring is no longer stretched or compressed. At this point, the potential energy is at its minimum (often taken as zero), and the mass is moving at its greatest speed. All of the energy that was previously stored in the spring has now been converted into kinetic energy. This means that through the formula Uk = 1/2(m)(v)^2, since velocity is maxed, the kinetic energy is maxed and equal to the system's total energy. On the other hand, through the formula Us = 1/2(k)(A)^2, since the block is at a displacement of 0, it has no potential energy.

As the mass continues past equilibrium toward the opposite side, the process reverses. The spring begins storing elastic potential energy again while the kinetic energy decreases. When the mass reaches the opposite maximum displacement, it once again momentarily stops, and all the energy is stored as potential energy. This continuous exchange between potential and kinetic energy allows the motion to repeat in a smooth and predictable cycle.

Energy in a Pendulum

At the highest points of a pendulum’s swing, the bob is momentarily at rest. Its speed is zero, so its kinetic energy is zero. However, it is at its greatest height relative to the lowest point, so its gravitational potential energy is at a maximum. As the pendulum swings downward toward equilibrium, gravitational potential energy decreases and kinetic energy increases. When the pendulum reaches the lowest point, its height is minimal, and its speed is greatest, meaning kinetic energy is at a maximum and gravitational potential energy is at a minimum. As it continues upward on the other side, the energy shifts back again.

An important idea in SHM is that the total energy of the system depends on amplitude. If you pull a spring back farther before releasing it, you store more potential energy in the spring. This means the system will have a greater maximum kinetic energy as it passes through equilibrium. However, even though the total energy increases with amplitude, the period of an ideal spring–mass system or small-angle pendulum does not depend on that energy. A system with a larger amplitude has more energy, but it still oscillates with the same period. This independence of period from energy is one of the defining characteristics of true simple harmonic motion.

Why Amplitude Affects Energy

Here’s a deeper conceptual point: The total energy of an SHM system depends on amplitude.

If you pull the spring back farther:

The maximum displacement increases

The maximum potential energy increases

The maximum kinetic energy increases

A larger amplitude in simple harmonic motion means the system carries more total energy, yet the period remains completely unchanged for ideal SHM. This leads to the interesting result that even though a bigger amplitude corresponds to greater energy, the oscillation still takes the same amount of time, whether it’s an ideal spring or a small‑angle pendulum. This counterintuitive fact often surprises students and highlights a defining feature of true SHM.

Energy Graph in SHM

Energy relationships can also be visualized graphically.

On this graph:

Potential energy increases as displacement increases (blue)

Kinetic energy decreases as displacement increases (green)

The total energy is a constant horizontal line (purple)

Real-World Example: Tuning Fork

A bungee jumper provides a clear real-world example of energy transfer in a spring system. When the jumper first steps off the platform, they begin with gravitational potential energy due to their height above the ground. As they fall, that gravitational potential energy is converted into kinetic energy, and their speed increases.

Once the bungee cord begins to stretch, something important happens. The cord starts storing elastic potential energy. As the cord stretches more and more, elastic potential energy increases while the jumper’s kinetic energy decreases. At the lowest point of the motion, the jumper momentarily comes to rest. At that instant, kinetic energy is zero, and the energy is stored primarily as elastic potential energy in the stretched cord.

After that lowest point, the cord pulls the jumper upward. Elastic potential energy converts back into kinetic energy as the jumper accelerates upward. As they rise higher, kinetic energy gradually converts back into gravitational potential energy. In an ideal system without air resistance, this energy would continue transferring between gravitational potential energy, elastic potential energy, and kinetic energy in a repeating cycle.

This example clearly shows how energy moves through different forms in oscillatory motion. The jumper does not keep bouncing because someone pushes it again; the motion continues because energy is being transferred back and forth between stored forms and motion. The restoring force provided by the stretched cord is what enables this continuous exchange of energy.

The Deeper Pattern

In simple harmonic motion, maximum displacement corresponds to maximum potential energy and zero kinetic energy. Equilibrium corresponds to maximum kinetic energy and minimum potential energy. Throughout the entire motion, the total mechanical energy remains constant in an ideal system. The restoring force enables this continuous exchange, converting stored energy into motion and motion back into stored energy again and again. That repeating energy cycle is what keeps SHM going.