5.5 Simple Harmonic Motion

Simple harmonic motion (SHM) describes the back-and-forth movement of an object around a central equilibrium point, where the restoring force is always proportional to how far the object has been displaced. It shows up constantly in physics, from spring-mass systems to pendulums to vibrating strings, and it's the foundation for understanding waves, sound, and even circuits later on.

Simple Harmonic Motion

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Application of Hooke's Law

Hooke's law is the starting point for SHM. It tells you that the restoring force on an object is proportional to its displacement from equilibrium and always points back toward that equilibrium position:

$F = -kx$

The negative sign is doing real work here. It means the force always opposes the displacement. Pull a spring to the right, and the force pulls left. Compress it to the left, and the force pushes right. That's what makes the object oscillate rather than just fly off in one direction.

The spring constant $k$ (measured in N/m) tells you how stiff the spring is. A large $k$ means a stiff spring that exerts a strong restoring force even for small displacements.

Energy in SHM is where things get interesting. As the mass oscillates, energy continuously converts between two forms:

• Elastic potential energy: $PE = \frac{1}{2}kx^2$, which is maximum at the extremes (where displacement is greatest and the mass momentarily stops)
• Kinetic energy: $KE = \frac{1}{2}mv^2$, which is maximum at the equilibrium position (where displacement is zero and speed is greatest)

The total mechanical energy stays constant throughout the motion:

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

where $A$ is the amplitude. At the extremes, all energy is potential. At equilibrium, all energy is kinetic. Everywhere in between, it's a mix of both.

Characteristics of Periodic Motion

Periodic motion is any motion that repeats itself after a fixed time interval. SHM is a specific type of periodic motion where the restoring force follows Hooke's law. Here are the key quantities you need to know:

• Amplitude $A$: the maximum displacement from equilibrium. A spring stretched 0.15 m from its rest position has an amplitude of 0.15 m.
• Period $T$: the time for one complete cycle (out and back). Measured in seconds.
• Frequency $f$: the number of complete cycles per second, measured in hertz (Hz). Related to period by $T = \frac{1}{f}$.
• Angular frequency $\omega$: the rate of oscillation in radians per second. Related to period and frequency by $\omega = \frac{2\pi}{T} = 2\pi f$.

A pendulum that swings back and forth once every 2.0 s has a period of 2.0 s, a frequency of 0.50 Hz, and an angular frequency of $\pi$ rad/s.

Oscillation and Phase

Oscillation refers to the repetitive back-and-forth variation about an equilibrium position. Phase describes where in its cycle an oscillating object currently is. Two objects oscillating with the same frequency and amplitude can still look different if they're out of phase, meaning they reach their maximum displacement at different times.

For SHM specifically, the position as a function of time can be written as $x(t) = A\cos(\omega t + \phi)$, where $\phi$ is the phase constant that sets the starting position of the oscillation at $t = 0$.

Simple Harmonic Oscillator Problems

Two classic SHM systems show up repeatedly in problems, and each has its own period formula.

Spring-mass system:

$T = 2\pi \sqrt{\frac{m}{k}}$

The period depends on mass $m$ and spring constant $k$. Notice that amplitude does not appear in this equation. A spring-mass system has the same period whether you pull it back 1 cm or 10 cm.

Example: A 0.50 kg mass is attached to a spring with $k = 200$ N/m. Find the period.

1. Identify givens: $m = 0.50$ kg, $k = 200$ N/m
2. Substitute into the formula: $T = 2\pi \sqrt{\frac{0.50}{200}}$
3. Calculate: $T = 2\pi \sqrt{0.0025} = 2\pi (0.050) \approx 0.31$ s

Simple pendulum:

$T = 2\pi \sqrt{\frac{L}{g}}$

The period depends on the length $L$ of the pendulum and gravitational acceleration $g$. Mass does not affect the period of a pendulum (a common exam trap). This formula also only holds for small angles of swing, typically less than about 15°.

Example: A pendulum is 0.80 m long. Find its period on Earth.

1. Identify givens: $L = 0.80$ m, $g = 9.81$ m/s²
2. Substitute: $T = 2\pi \sqrt{\frac{0.80}{9.81}}$
3. Calculate: $T = 2\pi \sqrt{0.0816} = 2\pi (0.286) \approx 1.80$ s

Comparing the two systems: Both exhibit SHM, but their restoring forces differ. For the spring-mass system, the restoring force comes from the spring itself. For the pendulum, it's the component of gravity along the arc. This is why changing mass affects the spring-mass period but not the pendulum period, and why changing gravity affects the pendulum period but not the spring-mass period.

Advanced Concepts

• Resonance occurs when an external periodic force drives an oscillating system at its natural frequency. The amplitude builds up dramatically because energy is being added in sync with the oscillation. This is why soldiers break step crossing a bridge, and why a singer can shatter a glass at the right pitch.
• Damping is the gradual loss of amplitude over time due to friction, air resistance, or other dissipative forces. In a real pendulum, for instance, each swing is slightly smaller than the last as mechanical energy converts to thermal energy.
• Real-world oscillating systems always involve some damping. Engineers have to account for both damping and resonance when designing structures, vehicles, and instruments.