Waves & oscillations

Oscillations

Here

Use springs and pendulums to connect equilibrium, sinusoidal motion, and energy exchange.

Resonance

Use coupled oscillators to connect driven resonance, normal modes, and the first hints of wave motion.

Waves

Build intuition for wave propagation by comparing what the pattern does with what the medium does.

Standing Waves and Harmonics

Finish

Connect reflected waves, nodes, antinodes, harmonics, and musical pitch on strings and air columns.

Oscillations

An oscillation is motion that repeats around an equilibrium point. Pull a mass on a spring, pluck a string, or nudge a pendulum and the system does not simply return once and stop. Inertia carries it past equilibrium, the restoring force pulls it back, and the cycle repeats. The cleanest version of that story is simple harmonic motion, where the restoring force is proportional to displacement.

Equilibrium and Restoring Force

A stable equilibrium is a point the system naturally returns to after a small disturbance. For a spring, Hooke’s law says

F = -kx.

In this case, $x=0$ is the equilibrium point of the system.
If the mass is displaced by $x$ , the spring pulls back with a force proportional to how far it has been stretched or compressed. The minus sign matters: it reminds us that the force points back toward equilibrium. This type of force is called a restoring force.

Period and Frequency

The period $T$ is the interval of time it takes to complete one full oscillation, or cycle, of the system. For an ideal mass-spring oscillator,

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

A stiffer spring makes the oscillation quicker, while a larger mass makes it slower. For small swings, a pendulum follows the same basic timing idea with

T_{\text{pendulum}} \approx 2\pi \sqrt{\frac{L}{g}},

so a longer pendulum swings more slowly than a shorter one. It’s also convenient to define frequency $f$ and angular frequency $\omega$ as

f = \frac{1}{T}, \qquad \omega = \frac{2\pi}{T}.

These measure cycles per second and radians per second, respectively (where each cycle corresponds to $2\pi$ radians). Note that “cycles” and “radians” are considered dimensionless.

A Sinusoidal Pattern

In ideal simple harmonic motion, position and velocity follow simple trigonometric functions. Here $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ sets the starting phase. Click either equation below to open a plot and see how those three parameters reshape the curve.

Due to the restoring force, the acceleration is proportional to displacement and directed back toward equilibrium:

a(t) = -\omega^2 x(t).

Energy Exchange

An oscillator is a continual trade between kinetic energy and stored potential energy. For a spring,

U_s = \frac{1}{2}kx^2, \qquad K = \frac{1}{2}mv^2, \qquad E_{\text{mech}} = K + U_s.

If damping is negligible, the total mechanical energy stays fixed and simply shifts form over the cycle. Real systems lose some of that mechanical energy to internal friction, air resistance, or sound, so the motion gradually fades unless something keeps driving it.

One Cycle

As the system oscillates, kinetic energy is continuously exchanged with spring potential energy. It's helpful to pause and consider the energy and motion at key points in each cycle.

t = 0

Right turning point

Released from rest

Displacement +A

Speed 0

Spring potential is largest.

t = T/4

Center crossing

Moving left fastest

Displacement 0

Speed maximum

Kinetic energy is largest.

t = T/2

Left turning point

About to reverse

Displacement -A

Speed 0

Spring potential is largest.

t = 3T/4

Center crossing

Moving right fastest

Displacement 0

Speed maximum

Kinetic energy is largest.

As you explore the simulation below, try this sequence:

Set damping to zero and compare how changing $m$ and $k$ changes the cycle time.
Watch the energy bars at the center crossing and at the turning points.
Raise the damping and notice how the amplitude envelope shrinks as mechanical energy leaves the system.

Spring-Mass Oscillator

Explore how amplitude, mass, spring stiffness, and damping shape the motion and energy flow.

Problem Solving

Find the Period and Frequency of a Spring

Problem

A mass-spring oscillator has mass $m = 0.50\,\text{kg}$ and spring constant $k = 200\,\text{N/m}$ . Find the period $T$ and frequency $f$ .

Given

$m = 0.50\,\text{kg}$
$k = 200\,\text{N/m}$

Use

For an ideal mass-spring oscillator,

T = 2\pi \sqrt{\frac{m}{k}}, \qquad f = \frac{1}{T}.

Solve

First compute the period:

T = 2\pi \sqrt{\frac{0.50}{200}} = 2\pi \sqrt{0.0025} = 2\pi(0.05) \approx 0.314\,\text{s}.

Now compute the frequency:

f = \frac{1}{0.314} \approx 3.18\,\text{Hz}.

Check

The result makes sense: a fairly stiff spring and a modest mass should oscillate quickly, so a short period and a frequency a little above $3\,\text{Hz}$ are both reasonable.

Use Energy to Find the Maximum Speed

Problem

A spring with constant $k = 80\,\text{N/m}$ oscillates with amplitude $A = 0.15\,\text{m}$ while attached to a mass $m = 0.40\,\text{kg}$ . Find the total mechanical energy and the maximum speed.

Given

$k = 80\,\text{N/m}$
$A = 0.15\,\text{m}$
$m = 0.40\,\text{kg}$

Use

At the turning point, all of the mechanical energy is spring potential energy:

E_{\text{mech}} = \frac{1}{2}kA^2.

At equilibrium, the speed is largest and all of the mechanical energy is kinetic:

E_{\text{mech}} = \frac{1}{2}mv_{\max}^2.

Solve

First find the total mechanical energy:

E_{\text{mech}} = \frac{1}{2}(80)(0.15^2) = 40(0.0225) = 0.90\,\text{J}.

Now set that equal to the maximum kinetic energy:

0.90 = \frac{1}{2}(0.40)v_{\max}^2 = 0.20v_{\max}^2.

v_{\max}^2 = \frac{0.90}{0.20} = 4.5 \qquad \Rightarrow \qquad v_{\max} \approx 2.12\,\text{m/s}.

Check

The energy stays the same the whole time, but its form changes. At the turning points, the oscillator stores the full $0.90\,\text{J}$ as spring potential energy. At equilibrium, that same $0.90\,\text{J}$ appears as kinetic energy, which is why the speed is largest there.

Oscillations Checkpoint

Question 1 of 3

A spring-mass oscillator is displaced farther from equilibrium. How does the restoring force change in ideal simple harmonic motion?

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