# Oscillations and Simple Harmonic Motion

Physics I · Principles of Mechanics · https://tryals.app/learn/physics-i/oscillations-and-simple-harmonic-motion

## Oscillations and SHM

Push almost anything slightly away from equilibrium, and the restoring force is proportional to displacement. That single fact makes the sine wave the universal motion of the gently disturbed world.

**Simple harmonic motion** (SHM) occurs when this restoring force acts:

$F = -kx \quad\Longrightarrow\quad \frac{d^2x}{dt^2} = -\omega^2 x$

where $\omega = \sqrt{k/m}$ is the **angular frequency**.

**General Solution**

$x(t) = A\cos(\omega t + \phi)$

| Parameter | Meaning |
| --- | --- |
| $A$ | Amplitude (max displacement) |
| $\omega$ | Angular frequency (rad/s) |
| $\phi$ | Phase constant |
| $T = 2\pi/\omega$ | Period |
| $f = 1/T$ | Frequency (Hz) |

**Physics Link:** SHM is the universal model for small oscillations about any stable equilibrium. Taylor-expand the potential energy to second order and you always get $F \approx -kx$.

## Energy and Oscillators

**Energy in SHM** oscillates between kinetic and potential, keeping total energy constant:

$E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$

At $x=0$, energy is all kinetic. At $x=\pm A$, energy is all potential.

**Common Oscillators**

| System | Period Formula |
| --- | --- |
| **Mass-spring** | $T = 2\pi\sqrt{m/k}$ |
| **Simple pendulum** | $T = 2\pi\sqrt{L/g}$ (for small angles $\theta \ll 1$) |

**Damped oscillations** add a friction term: $\ddot{x} + 2\gamma\dot{x} + \omega_0^{2} x = 0$, producing an exponentially decaying amplitude $Ae^{-\gamma t}$.

**Common Pitfall:** The period of a mass-spring oscillator does not depend on amplitude. A bigger swing travels farther but moves proportionally faster, and the two effects cancel exactly.

## Practice questions

2 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. A pendulum’s period is independent of its amplitude for small swings but not for large ones. Why does amplitude-independence fail at large angles?

A. The assumption that the string is massless
B. The approximation that $\sin\theta \approx \theta$
C. Conservation of energy, which fails for large oscillations
D. Air resistance, which becomes significant at large amplitudes

**Answer:** B. The approximation that $\sin\theta \approx \theta$

**Why:** SHM is defined by a restoring force proportional to displacement. The pendulum only approximates that condition, and amplitude-independence is a property of the approximation rather than of the pendulum.

Page: https://tryals.app/practice/physics-i/oscillations-and-simple-harmonic-motion/a-pendulums-period-is-independent-of-its-amplitude-for-small-swings

### 2. Order the phases of one full cycle of a mass on a spring (starting from maximum compression).

**Answer:**

1. Maximum compression ($x = -A$)
2. Passes through equilibrium moving right ($x = 0)$
3. Maximum extension (x = +A)
4. Passes through equilibrium moving left ($x = 0)$
5. Returns to maximum compression ($x = -A$)

**Why:** SHM is periodic: compression $\to$ equilibrium $\to$ extension $\to$ equilibrium $\to$ compression.

Page: https://tryals.app/practice/physics-i/oscillations-and-simple-harmonic-motion/order-the-phases-of-one-full-cycle-of-a-mass-on-a-spring-starting