Simple harmonic motion (SHM) is the most important type of oscillation in physics. A pendulum swinging, a mass bouncing on a spring, a vibrating guitar string, an alternating electric current — all are examples of simple harmonic motion or close approximations of it. Understanding SHM is the gateway to acoustics, optics, electrical circuits, quantum mechanics, and the physics of virtually every oscillating system you will encounter.
Simple harmonic motion is oscillatory motion in which the restoring force is directly proportional to the displacement from the equilibrium position and always directed toward it. Mathematically: F = −kx, where k is a positive constant (the stiffness) and x is displacement. The motion is sinusoidal: x(t) = A cos(ωt + φ).
What Makes Motion "Simple Harmonic"?
SHM has a precise definition: the restoring force must be proportional to displacement and opposite in direction. In equation form:
The negative sign is critical — it means the force always pushes or pulls the object back toward equilibrium (x = 0), not away from it. The greater the displacement, the stronger the restoring force. This produces the characteristic sinusoidal oscillation: the object overshoots equilibrium, the force reverses direction, it's pulled back, overshoots in the other direction, and the cycle repeats indefinitely (in the absence of damping).
Applying Newton's second law, F = ma = m(d²x/dt²), gives the equation of motion:
where ω = √(k/m) is the angular frequency (rad/s). The solution to this differential equation is:
where A is the amplitude (maximum displacement in metres), ω is angular frequency (rad/s), t is time (s), and φ is the initial phase (rad, determined by initial conditions).
Key SHM Quantities
| Quantity | Symbol | Formula | Unit |
|---|---|---|---|
| Amplitude | A | Maximum displacement from equilibrium | m |
| Period | T | T = 2π/ω = 1/f | s |
| Frequency | f | f = 1/T = ω/(2π) | Hz |
| Angular frequency | ω | ω = 2πf = √(k/m) | rad/s |
| Maximum velocity | v_max | v_max = Aω (at equilibrium) | m/s |
The Simple Pendulum
A simple pendulum — a mass m on a string of length L, swinging through small angles — is the classic SHM example. For small angles (θ < ~15°), the restoring force along the arc is approximately F = −mg sinθ ≈ −mgθ, which is proportional to displacement. This gives SHM with period:
where L is string length (m) and g = 9.8 m/s² is gravitational acceleration. Note what does not appear: mass. The period of a pendulum is independent of the mass of the bob. A 1 kg bob and a 10 kg bob on the same length string swing with identical periods. This was one of Galileo's great discoveries, made (allegedly) by watching a chandelier swing in Pisa Cathedral.
The period does depend on g — a pendulum beats slightly faster at the poles (where g is larger) than at the equator. This effect is significant enough to require correction in precision pendulum clocks used in different latitudes.
Worked example: pendulum period
A pendulum has length L = 1.00 m. Find its period on Earth (g = 9.8 m/s²).
A 1-metre pendulum has a period of almost exactly 2 seconds. This is why "seconds pendulums" — used in grandfather clocks — have lengths of about 1 metre.
Mass-Spring System
A mass m attached to a spring of spring constant k (where Hooke's law F = −kx applies) undergoes SHM with period:
Here, mass does matter: a heavier mass oscillates more slowly (larger T); a stiffer spring (larger k) oscillates faster (smaller T). A 0.5 kg mass on a spring with k = 200 N/m:
Energy in SHM
Energy continuously exchanges between kinetic and potential forms throughout SHM — a perfect illustration of conservation of energy.
At maximum displacement (amplitude A): velocity = 0 → KE = 0; PE = maximum = ½kA²
At equilibrium (x = 0): velocity = maximum (v_max = Aω) → KE = maximum = ½mω²A²; PE = 0
Total mechanical energy at any point:
This energy is independent of time — it depends only on amplitude A and spring constant k. Doubling the amplitude quadruples the energy (E ∝ A²).
A larger amplitude means the object travels further in each cycle — but it also moves faster (because it starts from a higher PE, converting more to KE). These two effects exactly cancel: larger distance but proportionally higher speed gives the same period. This amplitude-independence is a defining property of SHM and does not hold for large-angle pendulums or nonlinear oscillators.
Velocity and Acceleration in SHM
Velocity and acceleration as functions of time in SHM:
Velocity is 90° out of phase with displacement (maximum velocity when displacement is zero). Acceleration is 180° out of phase with displacement (maximum acceleration when displacement is maximum, directed oppositely). Acceleration is always proportional to displacement and directed toward equilibrium — the defining property of SHM.
Real-World Examples of SHM
Musical strings: A guitar string vibrates in SHM (approximately). The restoring force is the string tension. The frequency determines the pitch; the amplitude determines the loudness.
Quartz watches: A quartz crystal oscillates at 32,768 Hz under an applied voltage (piezoelectric effect). The circuit counts oscillations to keep time. The stability of the SHM frequency makes quartz clocks far more accurate than pendulum clocks.
LC circuits: An inductor (L) and capacitor (C) in a circuit exchange energy between magnetic field (inductor) and electric field (capacitor) in an exactly analogous way to a mass-spring system. The "natural frequency" is f = 1/(2π√LC) — the direct analogue of the spring formula.
Molecular vibrations: Atoms in molecules oscillate about their equilibrium bond lengths. For small displacements, the restoring force is approximately harmonic (F ≈ −kx), so molecular vibrations are approximately SHM. The frequencies, typically in the infrared range, are the basis of infrared spectroscopy.
Damped and Forced Oscillations
Real oscillators lose energy to friction and air resistance — they are damped. In underdamped systems, amplitude decreases exponentially with time while the frequency remains approximately the same. In overdamped systems, the object returns slowly to equilibrium without oscillating. In critically damped systems (the engineering ideal for shock absorbers), the object returns to equilibrium as quickly as possible without oscillating.
Resonance occurs when an oscillator is driven at its natural frequency. The amplitude grows — potentially to destructive levels. The Tacoma Narrows Bridge collapsed in 1940 partly due to wind-induced resonance. Microwave ovens exploit resonance: the 2.45 GHz microwaves are close to the natural rotational frequency of water molecules.
Frequently Asked Questions
What is simple harmonic motion?
Simple harmonic motion (SHM) is oscillatory motion where the restoring force is proportional to displacement and directed toward equilibrium: F = −kx. This produces sinusoidal motion: x(t) = A cos(ωt + φ). It is the basis of pendulums, springs, acoustic vibrations, and electrical oscillations.
What is the period of a simple pendulum?
T = 2π√(L/g), where L is the pendulum length and g = 9.8 m/s². Period depends on length and gravity but NOT on mass or amplitude (for small angles). A 1-metre pendulum on Earth has a period of approximately 2.0 seconds.
What is the period of a mass-spring system?
T = 2π√(m/k), where m is mass (kg) and k is spring constant (N/m). Unlike a pendulum, the period of a spring system depends on mass. A heavier mass oscillates more slowly; a stiffer spring oscillates faster. Period does not depend on amplitude in SHM.
Why is SHM amplitude-independent?
Larger amplitude means both greater distance and proportionally greater speed (from larger energy), so the two effects exactly cancel and period remains constant. This amplitude-independence holds only for true SHM (linear restoring force). A pendulum is only approximately SHM — for large angles, the period increases with amplitude.
What is the difference between frequency and period in SHM?
Period (T) is the time for one complete oscillation, measured in seconds. Frequency (f) is the number of complete oscillations per second, measured in hertz (Hz). They are reciprocals: f = 1/T. Angular frequency ω = 2πf = 2π/T, measured in rad/s.
What is resonance in SHM?
Resonance occurs when a periodic driving force is applied at the natural frequency of an oscillating system. Energy is transferred most efficiently at the natural frequency, causing amplitude to grow — potentially to destructive levels. Examples: Tacoma Narrows Bridge (wind resonance), microwave ovens (resonance with water molecule rotation), MRI scanners (nuclear magnetic resonance).
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Written by
Dr. Marcus WebbTheoretical physicist and science communicator with a PhD from Caltech. Research background in classical mechanics and gravitational physics. Passionate about making advanced physics accessible to all learners.
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