# Oscillations and Periodic Motion

Mathematics I · Physics · https://tryals.app/learn/mathematics-i/oscillations-and-periodic-motion

## Motion That Repeats

**Oscillation** is motion that repeats around an equilibrium, from pendulums to atoms. Its ideal form is **simple harmonic motion (SHM)**, which occurs when the restoring force is proportional to displacement: **Hooke's law**, $F = -kx$.

Combining Hooke's law with Newton's second law yields the differential equation $\ddot{x} = --\omega^2 x$, where angular frequency is $\omega = \sqrt{k/m}$. Its solution is a **sinusoid**, $x(t) = A\cos(\omega t + \phi)$, where $A$ is the maximum displacement.

| Property | Symbol | Formula / Meaning |
| --- | --- | --- |
| **Amplitude** | $A$ | Maximum displacement |
| **Period** | $T$ | Time per cycle, $\frac{2\pi}{\omega}$ |
| **Frequency** | $f$ | Cycles per second, $\frac{1}{T}$ |

## The Core Law & Real Limits

**Common pitfall:** Thinking a larger **amplitude** makes an ideal oscillator take **longer** per cycle. In SHM, the **period is independent of amplitude**: a bigger swing feels a proportionally bigger restoring force, covering extra distance in the same time.

Real oscillators experience **damping** (friction), which shrinks amplitude over time. When an oscillator is driven by an external periodic force matching its natural frequency $\omega$, **resonance** occurs: energy accumulates and the amplitude grows dramatically.

Resonance powers musical instruments and tuned circuits, but can destroy structures if left unchecked.

## Practice questions

5 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Doubling the amplitude of an ideal spring oscillator leaves its period unchanged.

**Answer:** True

**Why:** True. $T = 2\pi\sqrt{m/k}$ contains no amplitude term: a wider swing covers more distance at proportionally higher speed and takes the same time. This isochronism is what makes pendulum clocks possible, and it holds only while the restoring force stays proportional to displacement, real springs lose it when stretched far.

Page: https://tryals.app/practice/mathematics-i/oscillations-and-periodic-motion/doubling-the-amplitude-of-an-ideal-spring-oscillator-leaves-its

### 2. Order the derivation of simple harmonic motion from Hooke's law.

**Answer:**

1. Write Hooke's law: the restoring force is F = -kx
2. Substitute into Newton's second law to get m times the second derivative of x equal to -kx
3. Divide by m to reach the second derivative of x equal to -(k/m) x
4. Recognise the standard form with omega squared equal to k over m
5. Conclude the solution is the sinusoid x(t) = A cos(omega t + phi)

**Why:** This is the whole lesson in five lines: a linear restoring force plus the second law gives a second-order differential equation whose solutions are sinusoids. The minus sign is essential, it is what makes the force restoring, and without it the solutions would grow exponentially instead of oscillating.

Page: https://tryals.app/practice/mathematics-i/oscillations-and-periodic-motion/order-the-derivation-of-simple-harmonic-motion-from-hookes-law

### 3. An ideal spring oscillator has its amplitude doubled, with the same mass and spring. Sort each quantity by whether it changes.

**Answer:**

- Changes when the amplitude is doubled: The maximum displacement, The maximum speed, The total energy stored
- Unchanged when the amplitude is doubled: The period, The frequency, The angular frequency omega

**Why:** Since $\omega = \sqrt{k/m}$ contains no A, the period, frequency and angular frequency are all untouched, that is the defining and counterintuitive feature of simple harmonic motion. The displacement, the maximum speed (which is $A\omega$) and the stored energy (which goes as $A^{2}$) all depend on the amplitude.

Page: https://tryals.app/practice/mathematics-i/oscillations-and-periodic-motion/an-ideal-spring-oscillator-has-its-amplitude-doubled-with-the-same

### 4. A child on a swing pumps their legs in time with the motion and the amplitude grows enormously. Which principle is being exploited?

A. Conservation of momentum, transferred from the legs to the swing
B. Angular momentum conservation as the body's mass distribution changes
C. Resonance, since energy added at the natural frequency accumulates
D. Reduction of the effective pendulum length, which raises the frequency

**Answer:** C. Resonance, since energy added at the natural frequency accumulates

**Why:** Timing is everything: pushes delivered at the natural frequency add coherently, and mistimed ones partly undo the previous ones. The same principle tunes a radio and collapsed the Tacoma Narrows bridge, which is why soldiers break step crossing one.

Page: https://tryals.app/practice/mathematics-i/oscillations-and-periodic-motion/a-child-on-a-swing-pumps-their-legs-in-time-with-the-motion-and-the

### 5. Complete the statement about resonance.

**Answer:** Resonance occurs when an oscillator is driven by a periodic force at **its own natural frequency**, so that energy accumulates and the amplitude grows dramatically.

**Why:** Driving at the natural frequency means every push arrives in phase with the motion, so energy is added cycle after cycle. Off that frequency the pushes eventually fight the motion and the amplitude stays bounded, which is why soldiers break step on bridges.

Page: https://tryals.app/practice/mathematics-i/oscillations-and-periodic-motion/complete-the-statement-about-resonance