Harmonic Motion

Learning objectives

By the end of this module, the learner should be able to:

• explain oscillation about an equilibrium position;
• describe restoring force using Hooke's law;
• calculate angular frequency, period, and frequency;
• interpret displacement, velocity, and acceleration graphs;
• explain why mass and stiffness affect vibration behaviour.

First-principles explanation

Harmonic motion occurs when a system is displaced from equilibrium and experiences a restoring force that pulls it back. In an ideal spring-mass system, the restoring force is proportional to displacement and acts in the opposite direction.

The motion repeats because the system continually exchanges potential energy in the spring with kinetic energy in the moving mass.

Equations and units

• F is restoring force, measured in newtons (N).
• k is spring constant, measured in newtons per metre (N/m).
• x is displacement, measured in metres (m).
• m is mass, measured in kilograms (kg).
• ω is angular frequency, measured in radians per second (rad/s).
• T is period, measured in seconds (s).
• f is frequency, measured in hertz (Hz).

Worked example

Problem: A 1 kg mass is attached to a spring with stiffness 10 N/m. Find the angular frequency and period.

Known values:

• m = 1 kg
• k = 10 N/m

Calculation:

ω = √(k / m) = √(10 / 1) = 3.162 rad/s

T = 2π / ω = 2π / 3.162 = 1.987 s

Answer: The angular frequency is approximately 3.16 rad/s and the period is approximately 1.99 s.

Interactive simulator

The Harmonic Motion simulator lets the learner adjust amplitude, mass, and spring constant, then observe the effect on displacement, velocity, acceleration, period, frequency, and total energy.

Local launch command:

python -m streamlit run simulations/harmonic_motion/streamlit_web_app.py --server.port 8506

Engineering meaning

Harmonic motion appears in vehicle suspension, vibrating machinery, sensors, structures, and control systems. Engineers must understand oscillation because natural frequency, stiffness, and mass can determine whether a system behaves safely or dangerously.

Increasing stiffness raises the natural frequency. Increasing mass lowers it. This principle is central to vibration analysis and mechanical design.

Assumptions and limits

• The spring is ideal and obeys Hooke's law.
• There is no damping.
• There is no external driving force.
• Motion is one-dimensional.
• Mass and spring stiffness remain constant.

Challenge questions

1. What happens to the period if the mass is increased?
2. What happens to frequency if the spring constant is increased?
3. Why is velocity zero at maximum displacement?
4. Why is acceleration greatest when displacement is greatest?