Oscillations

Simple harmonic motion 简谐运动 (SHM) is a back-and-forth oscillation 振动 caused by a restoring force 回复力 that is proportional to the displacement from equilibrium and always points back toward it: $F=-kx$. A mass on a spring and (for small angles) a pendulum are the standard examples. Because the force grows with displacement, the motion is smooth and repeating, tracing a sine curve in time.

In SHM the acceleration always points back towards equilibrium, opposite the displacement

The test for SHM is exactly this: acceleration proportional to displacement and opposite in direction, $a=-\dfrac{k}{m}x$. A pendulum only obeys it for small swings, where $\sin\theta\approx\theta$; large swings are not quite SHM.

• The period 周期 $T$ is the time for one full cycle.
• The frequency 频率 $f=\dfrac{1}{T}$ is cycles per second (hertz).

For SHM these depend only on the system, not on the amplitude:

So a stiffer spring or smaller mass oscillates faster; a longer pendulum swings slower.

Worked example. A $0.25\ \text{kg}$ mass hangs on a spring of stiffness $k=100\ \text{N/m}$. Its period is

Worked example. A pendulum clock ticks with a period of exactly $2.0\ \text{s}$. How long is it? Rearranging $T=2\pi\sqrt{L/g}$,

Notice the amplitude never entered – a wide or narrow swing keeps the same time, which is what makes pendulums good clocks.

Tuning forks: a longer, heavier fork vibrates more slowly, giving a lower frequency

The displacement varies sinusoidally: $x(t)=A\cos(\omega t)$ (or sine), where $A$ is the amplitude 振幅 (maximum displacement) and $\omega=2\pi f$ is the angular frequency 角频率. Reading the motion:

Displacement varies sinusoidally with time in simple harmonic motion

• At the extremes ($x=\pm A$): displacement and restoring force are maximum, so acceleration is maximum, but velocity is zero.
• At equilibrium 平衡位置 ($x=0$): force and acceleration are zero, but speed is maximum.

Velocity and acceleration are also sinusoidal, shifted in phase 相位 from the displacement – velocity leads displacement by a quarter cycle, and acceleration is exactly opposite to displacement.

Displacement, velocity, and acceleration in SHM, each a quarter-cycle apart

Read the figure as a story: where $x$ is largest (the turning point), $v$ has just fallen to zero and $a$ is at its most negative, hauling the mass back; a quarter-cycle later the mass races through the middle at top speed with zero acceleration.

Energy sloshes between kinetic and potential while the total stays constant (no friction):

At the extremes it is all potential; at equilibrium it is all kinetic (maximum speed). Because $E\propto A^2$, doubling the amplitude quadruples the energy.

Kinetic and potential energy swap over a cycle while the total energy stays constant

Worked example. A $0.50\ \text{kg}$ mass on a spring of stiffness $k=200\ \text{N/m}$ oscillates with amplitude $A=0.10\ \text{m}$. Find its maximum speed. All the energy is kinetic at the equilibrium point, so $\tfrac12 kA^2=\tfrac12 mv_{\max}^2$:

The energy method is faster than tracking the sine functions when you only need the greatest speed.

• Test for SHM: the acceleration must be proportional to the displacement and directed back toward the middle ($a=-\tfrac{k}{m}x$).
• The period does not depend on amplitude — use $T=2\pi\sqrt{m/k}$ (spring) or $T=2\pi\sqrt{L/g}$ (pendulum, small angles).
• Speed is maximum at the middle (all kinetic) and zero at the extremes (all potential); acceleration is largest at the extremes.
• Use energy ($\tfrac12 kA^2 = \tfrac12 kx^2 + \tfrac12 mv^2$) to find the maximum speed quickly: $v_{\max}=A\sqrt{k/m}$.
• Total energy $\propto A^2$, so doubling the amplitude quadruples the energy.