Simple Harmonic Motion
The oscillation that results when a restoring force is proportional to displacement — the physics of pendulums, springs, and waves.
Definition and Conditions
A particle undergoes simple harmonic motion (SHM) when it experiences a restoring force proportional to its displacement from equilibrium and directed back toward it:
F = −kx
where k is a positive constant (the spring constant, in N/m) and x is displacement. The negative sign ensures the force always opposes displacement, creating oscillation. SHM is isochronous — the period is independent of amplitude (for small displacements).
Equations of Motion
Newton's second law F = ma with F = −kx gives the differential equation: mẍ + kx = 0, or equivalently ẍ + ω²x = 0, where ω = √(k/m) is the angular frequency (rad/s).
The general solution is:
- x(t) = A cos(ωt + φ) — displacement
- v(t) = −Aω sin(ωt + φ) — velocity
- a(t) = −Aω² cos(ωt + φ) — acceleration
where A is the amplitude and φ is the initial phase. The period T = 2π/ω and frequency f = 1/T = ω/2π.
Variable Table
| Symbol | Quantity | SI Unit |
|---|---|---|
| x | Displacement | m |
| A | Amplitude (max displacement) | m |
| ω | Angular frequency = √(k/m) | rad/s |
| T | Period = 2π/ω | s |
| f | Frequency = 1/T | Hz |
| k | Spring constant | N/m |
Energy in SHM
The total mechanical energy in SHM is conserved: E = ½kA² = constant. At any displacement x:
- Potential energy: PE = ½kx²
- Kinetic energy: KE = ½k(A² − x²) = ½mv²
- Total: E = KE + PE = ½kA²
Maximum speed occurs at x = 0: v_max = Aω. Maximum acceleration occurs at x = ±A: a_max = Aω².
Common SHM Systems
Mass-Spring System
For a mass m on a spring of constant k: ω = √(k/m) and T = 2π√(m/k). This applies to both horizontal and vertical configurations (gravity only shifts the equilibrium position, it does not change the period).
Simple Pendulum
For small angles (θ ≪ 1 radian), the restoring force is mg sin θ ≈ mgθ, giving SHM with ω = √(g/L) and T = 2π√(L/g). The period is independent of mass and amplitude. For larger angles, the period increases — a 15° swing is about 0.5% longer than predicted by SHM.
Physical Pendulum
An extended object pivoting about a fixed axis: T = 2π√(I/mgd), where I is the moment of inertia about the pivot and d is the distance from pivot to centre of mass.
Worked Examples
Example 1 — Mass-spring period
A 0.4 kg mass is attached to a spring with k = 100 N/m. Find the period and maximum speed when amplitude A = 0.05 m.
Solution: ω = √(100/0.4) = √250 ≈ 15.81 rad/s. T = 2π/ω = 2π/15.81 ≈ 0.397 s. v_max = Aω = 0.05 × 15.81 ≈ 0.79 m/s.
Example 2 — Pendulum length from period
A pendulum on Earth has a period of 2.0 s. Find its length.
Solution: T = 2π√(L/g) → L = g(T/2π)² = 9.81 × (2/2π)² = 9.81 × (1/π)² ≈ 0.993 m (the "seconds pendulum").
Example 3 — Energy in SHM
A 0.2 kg mass on a spring (k = 50 N/m) oscillates with amplitude 0.1 m. Find the speed at x = 0.06 m.
Solution: E = ½kA² = ½(50)(0.01) = 0.25 J. PE at x = 0.06: ½(50)(0.0036) = 0.09 J. KE = 0.25 − 0.09 = 0.16 J. v = √(2KE/m) = √(0.32/0.2) = √1.6 ≈ 1.26 m/s.
Common Mistakes
- Using the small-angle formula outside its range. T = 2π√(L/g) applies for θ ≲ 15°. For larger amplitudes, use the full elliptic-integral formula.
- Confusing period and frequency. f = 1/T and ω = 2πf. These are not interchangeable.
- Ignoring the ½ factor in energy expressions. E = ½kA² ≠ kA².
- Forgetting that a vertical spring still gives T = 2π√(m/k). Gravity shifts equilibrium but does not change the period because the net restoring force still obeys F = −kx measured from the new equilibrium.
Applications
- Timekeeping: Pendulum clocks, quartz oscillators, and atomic clocks all rely on highly stable periodic oscillations.
- Seismology: Seismographs use suspended mass systems sensitive to earth vibrations.
- Musical instruments: Strings, air columns, and membranes vibrate in SHM to produce sound.
- Engineering: Vibration isolation, suspension design, and resonance avoidance in bridges and buildings.
- Atomic physics: Molecular vibrations modelled as coupled harmonic oscillators determine infrared spectra.
Damped and Forced Oscillations
Real oscillators lose energy over time — this is called damped oscillation. The damping force is typically proportional to velocity: F_damp = −bv, giving the equation mẍ + bẋ + kx = 0. Depending on the damping constant b relative to the critical value b_c = 2√(km):
- Underdamped (b < b_c): oscillations decay exponentially — x(t) = Ae^(−γt) cos(ω′t + φ), where γ = b/2m
- Critically damped (b = b_c): returns to equilibrium fastest without oscillating
- Overdamped (b > b_c): slow exponential return to equilibrium
Forced oscillations occur when a periodic external force drives the system. When the driving frequency matches the natural frequency ω₀ = √(k/m), resonance occurs and amplitude grows dramatically. Engineers must avoid resonance in bridges, buildings, and aircraft structures — the Tacoma Narrows Bridge collapse (1940) is the classic cautionary example.
Related Topics
Frequently Asked Questions
What makes motion simple harmonic?
A restoring force proportional to displacement: F = −kx. This produces sinusoidal oscillations at fixed frequency, independent of amplitude.
Period of a simple pendulum?
T = 2π√(L/g). Independent of mass and amplitude (for small angles). A 1-metre pendulum on Earth has period ≈ 2 seconds.
Period of a mass-spring system?
T = 2π√(m/k). Stiffer spring → shorter period; heavier mass → longer period. Gravity doesn't change it.
Energy distribution in SHM?
Total E = ½kA² is constant. At x = 0 all energy is kinetic; at x = ±A all energy is potential. KE + PE = ½kA² always.
References
- Halliday, D., Resnick, R., & Krane, K. S. (2001). Physics (5th ed.). Wiley. Chapter 15.
- French, A. P. (1971). Vibrations and Waves. MIT Introductory Physics Series. Chapter 1–3.
- Crawford, F. S. (1968). Waves. Berkeley Physics Course Vol. 3. McGraw-Hill. Chapter 1.
- Kiran T. Bhansali. (2019). "The Simple Pendulum." American Journal of Physics, 87(2), 144–146.