Oscillations, Springs, and Pendulums

A swing, a vibrating ruler, and a mass on a spring all repeat motion around an equilibrium position. The details differ, but period, frequency, amplitude, and restoring force organize every case.

This lesson supports Chapter 13 of High School Physics.

Period and frequency describe repetition

The period \(T\) is the time for one cycle. Frequency \(f\) is the number of cycles per second, and \[f=\frac1T.\] The hertz is one cycle per second. Amplitude measures the largest displacement from equilibrium.

A spring supplies a restoring force

For an ideal horizontal spring within its linear elastic range, with displacement measured from its unstretched equilibrium, Hooke’s law is \(F_s=-kx\). The minus sign shows that the force points toward equilibrium. With negligible damping and spring mass, this mass-spring system has period \[T=2\pi\sqrt{\frac{m}{k}}.\] More mass increases the period, while a stiffer spring decreases it.

A simple pendulum depends mainly on length

For small angles, \(T=2\pi\sqrt{\frac{L}{g}}\). The bob’s mass does not appear in the formula. A longer pendulum swings more slowly. Large angles and air drag make the simple model less accurate.

Worked example

A vibration has period \(0.25\text{ s}\). Its frequency is \(f=\frac1T=\frac1{0.25}=4.0\text{ Hz}\). Four complete cycles occur each second.

Watch the idea in action

Professor Dave Explains gives a focused explanation of the chapter’s central idea. Pause before each calculation and predict the next step on paper.

Practice questions

1. What is a period?
2. What is frequency?
3. How are period and frequency related?
4. State Hooke’s law.
5. What happens to a spring oscillator’s period when mass increases?
6. Does pendulum period depend on bob mass in the small-angle model?

Answers

1. The time for one cycle.
2. The number of cycles per second.
3. \(f=\frac1T\).
4. \(F_s=-kx\).
5. It increases.
6. No.

Keep studying

High School Physics includes 12 full-length practice tests: two in the book and ten online.

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