Oscillations — SHM, Damped & Forced Oscillations

Oscillations in physics are repeated motions or repeated changes around an equilibrium position. If you want to understand the topic quickly, separate three cases: ideal simple harmonic motion, oscillations that lose energy, and oscillations driven by an external periodic force.

A mass on a spring, a small-angle pendulum, and an AC circuit can all oscillate. The fast way to sort them is this:

• SHM is the ideal case where the restoring effect is proportional to displacement.
• Damped oscillation means energy is being lost, so the amplitude shrinks with time.
• Forced oscillation means an external periodic input keeps driving the system.

If the driving frequency is close to the system's natural frequency, the response can become much larger. That basic effect is resonance.

What Makes a System Oscillate?

An oscillating system has two ingredients: an equilibrium position and a restoring effect that pushes the system back after it is displaced. Once the system moves back through equilibrium, inertia usually carries it past the center, so the motion repeats.

That repeating motion does not automatically mean SHM. SHM is a narrower, ideal model with a specific condition:

$$F = -kx$$

for a spring, or more generally a restoring effect proportional to displacement. The minus sign matters because it shows the force points back toward equilibrium.

Simple Harmonic Motion: The Ideal Case

In ideal simple harmonic motion, the acceleration is proportional to displacement and opposite in direction:

$$a = -\omega^2 x$$

That condition leads to sinusoidal motion. For a mass $m$ on a spring with spring constant $k$,

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

and

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

where $T$ is the period.

The useful intuition is simple: a stiffer spring pulls harder, so the oscillation is faster. A larger mass resists acceleration more, so the oscillation is slower.

Damped Oscillations: Why the Amplitude Shrinks

Real systems usually lose energy. Air resistance, friction, internal deformation, and electrical resistance all act as damping.

When damping matters, the motion still oscillates for a while, but the amplitude gets smaller with time. The system is not gaining enough energy to keep the original size of the motion.

For light damping, the motion still looks roughly periodic. For heavy damping, the system may return to equilibrium without completing repeated oscillations.

Forced Oscillations and Resonance

A forced oscillation happens when an external periodic influence keeps pushing the system. A child pumping a swing, a speaker cone driven by an alternating signal, or a building shaken by repeated ground motion are all examples.

The key point is that the driving frequency matters. If it is far from the natural frequency, the response may stay modest. If it is close, the amplitude can become much larger.

That large-response region is called resonance. To stay precise, the strongest response is often near the natural frequency for light damping, and the exact peak depends on damping and on what quantity you track.

Worked Example: One Spring, Three Ideas

Suppose a mass of $0.50\ \mathrm{kg}$ is attached to an ideal spring with $k = 200\ \mathrm{N/m}$.

First find the angular frequency:

$$\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{200}{0.50}} = \sqrt{400} = 20\ \mathrm{rad/s}$$

Now find the period:

$$T = \frac{2\pi}{\omega} = \frac{2\pi}{20} \approx 0.314\ \mathrm{s}$$

So in the ideal SHM model, the system completes one oscillation in about $0.314$ seconds. Its natural frequency is

$$f = \frac{\omega}{2\pi} = \frac{20}{2\pi} \approx 3.18\ \mathrm{Hz}$$

which means about $3.18$ oscillations each second.

Now use the same system to see the bigger picture:

• If you ignore friction and just release it, the motion is ideal SHM.
• If there is air resistance or internal friction, the amplitude gradually decreases, so the motion is damped.
• If you keep pushing it periodically with a motor or an external force, the motion is forced.

If that driving force repeats at a rate close to about $3.18\ \mathrm{Hz}$, and damping is light, the response can grow much larger than it would away from resonance.

This one example is enough to organize the topic: SHM describes the ideal timing, damping explains why real motion fades, and forcing explains how an outside input can sustain or enlarge the motion.

Common Mistakes in Oscillations

Calling every oscillation SHM

Back-and-forth motion alone is not enough. SHM requires a restoring effect proportional to displacement.

Thinking damping only changes the amplitude

For light damping, the biggest visible change is usually the shrinking amplitude, but damping also changes the detailed motion. It is not just a visual effect.

Assuming forced motion always grows without limit

Real systems usually have damping, and that limits the steady response. Without that point, resonance is easy to misunderstand.

Saying resonance must be exactly at the natural frequency in every case

That is too loose. In introductory physics, "near the natural frequency" is the safer statement unless the model and the measured quantity are specified.

Where Oscillations Show Up in Physics

Oscillation models appear in mechanical systems, sound and vibration, electrical circuits, molecular motion, clocks, sensors, and structural engineering. They matter because many real systems repeat, store energy, lose energy, and respond strongly to repeated driving.

That is why the same ideas show up in very different places: a car suspension, a pendulum clock, a guitar string, and an RLC circuit all use the same basic language of natural frequency, damping, and driving.

Try a Similar Problem

Take the same spring and double the mass to $1.0\ \mathrm{kg}$. Recompute $T$ and the natural frequency, then compare them with the original values. After that, ask what happens if a periodic driving force acts near the new natural frequency. If you want to go further, try solving the same question for a pendulum or an RLC circuit.