Simple Harmonic Motion
Simple Harmonic Motion (SHM) is a type of periodic oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and always directed toward it. The motion follows a sinusoidal pattern over time, characterized by constant amplitude, frequency, and period in the absence of damping. SHM is the basis for understanding pendulums, springs, sound waves, and alternating electric circuits.
Key Formula
x(t) = A·cos(ωt + φ), a = −ω²x
LaTeX: x(t) = A\cos(\omega t + \phi), \quad a = -\omega^2 x
| Symbol | Meaning | Unit |
|---|---|---|
| x | Displacement from equilibrium | m |
| A | Amplitude (maximum displacement) | m |
| \omega | Angular frequency | rad/s |
| t | Time | s |
| \phi | Initial phase angle | rad |
| a | Acceleration | m/s² |
Worked Example
Problem
An object undergoes SHM with amplitude A = 0.10 m and angular frequency ω = 5.0 rad/s. Find its maximum speed, maximum acceleration, and period.
Solution
Step 1: Maximum speed — vmax = Aω = 0.10 × 5.0 = 0.50 m/s. Step 2: Maximum acceleration — amax = Aω² = 0.10 × 25 = 2.5 m/s². Step 3: Period — T = 2π/ω = 2π/5.0 ≈ 1.257 s.
Answer
vmax = 0.50 m/s, amax = 2.5 m/s², T ≈ 1.26 s.
Displacement, velocity, and acceleration at key points in SHM
| Position | Displacement | Speed | Acceleration | Energy |
|---|---|---|---|---|
| At equilibrium | 0 | Maximum (Aω) | 0 | All kinetic |
| At +A (max positive) | A | 0 | −Aω² (max negative) | All potential |
| At −A (max negative) | −A | 0 | +Aω² (max positive) | All potential |
| At x = A/2 | A/2 | (√3/2)Aω | −Aω²/2 | Mixed |
| At x = −A/2 | −A/2 | (√3/2)Aω | +Aω²/2 | Mixed |
Interactive Tools
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Related Terms
Pendulum
A pendulum is a mass (called a bob) suspended from a fixed point by a string or rod that oscillates back and forth under the influence of gravity. For small angular displacements (less than about 15°), a simple pendulum exhibits simple harmonic motion, and its period depends only on its length and the local gravitational acceleration, not on mass or amplitude. Pendulums have historically been used in clocks and are fundamental to understanding oscillatory systems.
Spring-Mass System
A spring-mass system consists of a mass attached to an ideal spring that obeys Hooke's Law, where the restoring force is proportional to the displacement from equilibrium. When displaced and released, the mass oscillates in simple harmonic motion with a period that depends on the mass and the spring constant, but not on amplitude. This system is the canonical model for oscillatory behavior in physics and engineering.
Mechanical Resonance
Mechanical resonance occurs when an oscillating system is driven at its natural frequency, causing the amplitude of oscillation to grow dramatically — theoretically without bound in an undamped system. Every mechanical system has one or more natural frequencies at which it vibrates freely after being disturbed. Resonance is critical in engineering design, as it can cause catastrophic structural failure (Tacoma Narrows Bridge, 1940) or be harnessed usefully in musical instruments, clocks, and sensors.
The word "harmonic" derives from Greek "harmonikos" (musical, harmonious), reflecting the musical origins of oscillation studies. "Simple" distinguishes it from compound or damped oscillations. The mathematical framework was developed by Robert Hooke (1660) and fully formalized by Newton.