16.5 Energy and the Simple Harmonic Oscillator

Energy in Simple Harmonic Oscillators

Maximum velocity in harmonic oscillators

In a simple harmonic oscillator, the object moves fastest as it passes through the equilibrium position, where displacement is zero. At that point, all the system's energy is kinetic, so that's where you find the maximum velocity.

The formula for maximum velocity is:

$v_{max} = \omega A$

where $\omega$ is the angular frequency and $A$ is the amplitude (the maximum displacement from equilibrium). Angular frequency is determined by the system's physical properties:

$\omega = \sqrt{\frac{k}{m}}$

• $k$ is the spring constant, which measures how stiff the spring is. A stiffer spring means a larger $k$.
• $m$ is the mass of the oscillating object. A heavier object oscillates more slowly.

Combining these gives you a single useful equation:

$v_{max} = A\sqrt{\frac{k}{m}}$

Notice what this tells you: maximum velocity increases with a stiffer spring (higher $k$), a larger amplitude (higher $A$), or a lighter mass (lower $m$).

Energy transformations during oscillation

The total mechanical energy of a simple harmonic oscillator stays constant throughout the motion. At every point in the cycle, the sum of kinetic energy (KE) and potential energy (PE) equals the same total:

$E_{total} = KE + PE$

Energy continuously converts back and forth between these two forms:

• At the equilibrium position (displacement = 0): KE is at its maximum and PE is zero. The object is moving at its fastest, and the spring is neither stretched nor compressed.
• At maximum displacement (displacement = $\pm A$): PE is at its maximum and KE is zero. The object is momentarily at rest before reversing direction, and all the energy is stored in the spring.

Because energy is conserved, the maximum KE and maximum PE are equal to each other, and both equal the total energy of the system:

$E_{total} = KE_{max} = PE_{max} = \frac{1}{2}kA^2$

This is a key result. The total energy depends only on the spring constant and the amplitude squared. It does not depend on the mass. (Mass affects how fast the object moves, but the energy stored in the system is set by $k$ and $A$.)

Comparing Simple Harmonic Oscillators

Velocity factors in springs vs pendulums

Springs and pendulums are both simple harmonic oscillators, but different physical quantities control their behavior.

Spring-mass systems use the formula:

$v_{max} = A\sqrt{\frac{k}{m}}$

• Higher $k$ (stiffer spring) or larger $A$ increases $v_{max}$
• Larger $m$ (heavier object) decreases $v_{max}$
• Doubling $k$ increases $v_{max}$ by a factor of $\sqrt{2}$, while doubling $A$ doubles $v_{max}$

Pendulums (for small-angle oscillations) use:

$v_{max} = A\sqrt{\frac{g}{L}}$

• $L$ is the length of the pendulum and $g$ is gravitational acceleration
• A shorter pendulum or stronger gravitational field increases $v_{max}$
• Larger $A$ increases $v_{max}$, just like with springs

The parallel structure of these two equations is worth noticing. In both cases, $v_{max}$ is directly proportional to amplitude. The ratio under the square root captures the restoring force: $k/m$ for springs, $g/L$ for pendulums. Stronger restoring forces produce higher maximum velocities.

Characteristics of Simple Harmonic Motion

These terms come up frequently when describing oscillatory systems:

• Period ($T$): The time for one complete back-and-forth cycle.
• Displacement: How far the object is from its equilibrium position at any moment.
• Oscillation: The repetitive back-and-forth motion around an equilibrium position.
• Resonance: When a system is driven at a frequency matching its natural frequency, the amplitude of oscillation grows significantly. This is why a tuning fork vibrates strongly at a specific pitch.
• Damping: The gradual decrease in amplitude over time due to energy loss (from friction, air resistance, etc.). Real oscillators always experience some damping, which is why a swinging pendulum eventually stops.