Simple Harmonic Motion

What is Simple Harmonic Motion?

Simple Harmonic Motion (SHM) is a specific type of periodic motion where a restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.

It is the mathematical foundation for understanding everything from pendulum clocks and car suspensions to the vibrations of atoms in a crystal lattice.

Learning Goals: By the end of this guide, you should be able to:

1. Define SHM and its defining conditions.
2. Relate displacement, velocity, and acceleration during an oscillation.
3. Calculate the period of a mass-spring system and a simple pendulum.

The Defining Condition of SHM

For an object to be undergoing Simple Harmonic Motion, its acceleration ($a$) must be directly proportional to its displacement ($x$) from equilibrium, and directed strictly toward the equilibrium point.

Mathematically, this is expressed as:

Where:

• $a$ is the acceleration ($\text{m/s}^2$).
• $x$ is the displacement from equilibrium ($\text{m}$).
• $\omega$ is the angular frequency ($\text{rad/s}$), which is a constant for the system.
• The negative sign indicates that acceleration is always opposite to the displacement (it's a restoring acceleration).

By Newton's Second Law ($F = ma$), this also means the restoring force is proportional to displacement: $F = -kx$ (which is Hooke's Law for springs).

Kinematics of SHM

As an object oscillates, its position, velocity, and acceleration continuously change in a sinusoidal pattern.

Let's assume the object starts at maximum positive displacement ($x = A$) at $t = 0$.

1. Displacement ($x$): $x = A \cos(\omega t)$
2. Velocity ($v$): $v = -A\omega \sin(\omega t)$
   • Velocity is maximum at the equilibrium position ($x=0$).
   • Velocity is zero at the maximum displacement (the turning points).
3. Acceleration ($a$): $a = -A\omega^2 \cos(\omega t)$
   • Acceleration is maximum at the maximum displacement (where the restoring force is greatest).
   • Acceleration is zero at the equilibrium position.

Two Classic SHM Systems

1. The Mass-Spring System

A mass $m$ attached to a spring with spring constant $k$. The period $T$ (time for one full oscillation) depends only on the mass and the stiffness of the spring. It does not depend on the amplitude.

• A heavier mass means a longer period (slower oscillation).
• A stiffer spring (larger $k$) means a shorter period (faster oscillation).

2. The Simple Pendulum

A point mass on a string of length $L$, swinging through a small angle (usually less than $10^\circ$). The period $T$ depends only on the length of the pendulum and the local gravity. It does not depend on the mass of the bob or the amplitude of the swing.

• A longer string means a longer period.
• Pendulums swing slightly slower on the Moon because $g$ is smaller.

Energy in SHM

In an ideal SHM system with no friction, the total mechanical energy is conserved. The energy continuously transforms between Kinetic Energy (KE) and Potential Energy (PE).

• At maximum displacement ($x = \pm A$): The object is momentarily stationary. $\text{KE} = 0$, and $\text{PE}$ is at its maximum.
• At equilibrium ($x = 0$): The spring is un-stretched (or the pendulum is at its lowest point). $\text{PE} = 0$, and the object is moving at its maximum speed, so $\text{KE}$ is at its maximum.

Common Mistakes

1. Assuming period depends on amplitude: A pendulum pulled back $5^\circ$ takes exactly the same amount of time to swing as one pulled back $2^\circ$. Amplitude does not affect the period in ideal SHM.
2. Confusing velocity and acceleration maxima: Many students think that because speed is highest at the center, acceleration must be highest there too. It's the exact opposite! At the center, the spring is relaxed, so force and acceleration are zero, even though speed is at its peak.
3. Using large angles for pendulums: The formula $T = 2\pi\sqrt{L/g}$ is an approximation that only works for angles $< 10^\circ$. For larger swings, the motion is no longer purely "Simple" Harmonic.

Related Topics

• Mechanical Waves — A wave is essentially energy propagating through a medium composed of countless coupled SHM oscillators.
• Motion Graphs — The graphs of displacement, velocity, and acceleration in SHM are beautiful, interconnected sine and cosine waves.