14.2 Energy in Simple Harmonic Motion

Energy in Simple Harmonic Motion

Simple harmonic motion involves a continuous exchange of energy between kinetic and potential forms. As an object oscillates, its energy shifts back and forth while the total energy stays constant in ideal (frictionless) systems. Understanding this energy exchange is central to analyzing oscillatory motion.

The total energy in SHM depends on the amplitude of oscillation. Because energy scales with the square of the amplitude, doubling the amplitude quadruples the energy.

Kinetic and Potential Energy Interchange

At any point during oscillation, energy exists as some combination of kinetic energy (energy of motion) and potential energy (energy stored due to position). These two forms trade off throughout each cycle:

• At the extremes of oscillation (maximum displacement, $x = \pm A$), the object momentarily stops. Velocity is zero, so all the energy is potential: $PE = \frac{1}{2}kA^2$.
• At the equilibrium position ($x = 0$), the spring is at its natural length and stores no potential energy. The object moves at its maximum speed, so all the energy is kinetic: $KE = \frac{1}{2}mv_{max}^2$.
• Between these points, energy is split between both forms. Kinetic and potential energy each vary sinusoidally over time, and they're 90 degrees out of phase with each other. When one is at a maximum, the other is zero.

Total Energy Calculation in SHM

The total mechanical energy of an ideal SHM system is constant and can be written as:

$E = \frac{1}{2}kA^2$

where $k$ is the spring constant and $A$ is the amplitude. This expression works because at maximum displacement the energy is entirely potential, giving you the total in one clean formula.

At any arbitrary position $x$, energy conservation tells you:

$E = KE + PE = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$

Since $E$ is constant, you can solve for the speed at any position:

1. Start with $\frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$

2. Rearrange to get $v = \sqrt{\frac{k}{m}(A^2 - x^2)}$

3. Notice that at $x = 0$, this gives the maximum speed: $v_{max} = A\sqrt{\frac{k}{m}} = A\omega$

This connects the total energy to either the amplitude or the maximum velocity: $E = \frac{1}{2}kA^2 = \frac{1}{2}mv_{max}^2$.

Amplitude and Energy Relationship

Because $E = \frac{1}{2}kA^2$, energy is proportional to $A^2$. This has a few practical consequences:

• Doubling the amplitude quadruples the total energy (since $2^2 = 4$).
• Tripling the amplitude increases the energy by a factor of 9.
• A graph of energy vs. amplitude is a parabola opening upward.

A larger amplitude means greater maximum displacement and greater maximum speed, so both the peak potential energy and peak kinetic energy increase.

Energy Conservation in Oscillations

Real systems lose energy over time due to friction or air resistance. Here's how different scenarios compare:

• Undamped oscillations: Total energy stays constant. The amplitude never changes. This is the ideal case.
• Damped oscillations: Friction or drag converts mechanical energy into thermal energy. The amplitude decreases exponentially over time, and the total mechanical energy gradually drops.
• Driven oscillations: An external periodic force pumps energy into the system. In the steady state, the energy input per cycle from the driving force exactly balances the energy lost to damping, so the amplitude stabilizes. If the driving frequency matches the system's natural frequency ($\omega_{drive} = \omega_0$), resonance occurs and the amplitude reaches its maximum value.