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Simple Harmonic Motion
Learn the meaning of SHM, its important equations, motion of a spring and pendulum, energy changes, graphs, and exam-friendly examples.
Many objects move repeatedly around a central position: a mass attached to a spring moves back and forth, a pendulum swings from side to side, and a guitar string vibrates. This repeated motion is called oscillatory motion. A special type of oscillatory motion is called simple harmonic motion, or SHM.
Simple harmonic motion is oscillatory motion in which the restoring force, or acceleration, is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.
F ∝ −x and a ∝ −x
The minus sign shows that the force or acceleration always points back toward the equilibrium position.
1. Basic Ideas You Need to Know
Equilibrium position
The central position where the net force is zero. It is usually chosen as x = 0.
Displacement
The signed distance of the object from equilibrium at a particular instant. It is represented by x.
Amplitude
The maximum displacement from equilibrium. It is represented by A, so the object moves between +A and −A.
One complete oscillation
One complete to-and-fro motion that returns the object to its starting position and direction of motion.
Period, frequency and angular frequency
The period, T, is the time taken for one complete oscillation. Its SI unit is the second (s).
The frequency, f, is the number of complete oscillations per second. Its SI unit is the hertz (Hz), where 1 Hz = 1 s−1.
f = 1/T and T = 1/f
The angular frequency, ω, measures how quickly the phase of the motion changes. Its unit is radians per second (rad s−1).
ω = 2πf = 2π/T
2. The Restoring Force and Hooke’s Law
When an object is moved away from equilibrium, a force often acts to bring it back. This is called a restoring force. For an ideal spring, the restoring force follows Hooke’s law:
F = −kx
Here, k is the spring constant. A larger value of k means the spring is stiffer and produces more force for the same displacement.
Using Newton’s second law, F = ma:
ma = −kx ⇒ a = −(k/m)x
This equation has the SHM form because acceleration is proportional to displacement and opposite to it.
Real systems may lose energy because of friction or air resistance. The ideal SHM equations describe motion most accurately when damping is negligible and the restoring force remains proportional to displacement.
3. Position, Velocity and Acceleration in SHM
The position of an ideal oscillator can be written using a sine or cosine function. A general form is:
x(t) = A cos(ωt + φ)
The symbol φ is the phase constant, measured in radians. It tells us the starting condition of the motion. A sine form is equally valid because sine and cosine differ only by a phase shift.
Velocity
v(t) = −Aω sin(ωt + φ)
The maximum speed is:
vmax = Aω
The object is momentarily at rest at the extreme positions, where x = +A or x = −A. Its speed is greatest as it passes through equilibrium.
Acceleration
a(t) = −Aω2 cos(ωt + φ)
Because x(t) = A cos(ωt + φ), this can also be written as:
a = −ω2x
The maximum acceleration is:
amax = Aω2
- At equilibrium: x = 0, acceleration is zero, and speed is maximum.
- At either extreme: x = ±A, speed is zero, and the magnitude of acceleration is maximum.
- Direction: acceleration always points toward equilibrium, opposite to the displacement.
4. Mass–Spring System
A mass attached to an ideal spring on a frictionless surface is the standard example of SHM. The spring force is F = −kx. Combining this with Newton’s second law gives the angular frequency:
ω = √(k/m)
Therefore, the period and frequency are:
T = 2π√(m/k)
f = (1/2π)√(k/m)
- Increasing the mass m increases the period.
- Increasing the spring constant k decreases the period.
- For an ideal spring–mass oscillator, the period does not depend on amplitude.
5. Simple Pendulum as an Approximate SHM System
A simple pendulum consists of a small bob suspended by a light, inextensible string. For small angular displacements, its motion is approximately simple harmonic.
T = 2π√(L/g)
Here, L is the pendulum length in metres and g is the acceleration due to gravity in m s−2.
This formula assumes a small angle, a light string, a concentrated bob, and negligible air resistance and friction. For larger angles, the motion is still periodic, but it is not described exactly by the simple small-angle SHM formula.
6. Energy in Simple Harmonic Motion
In an ideal oscillator, mechanical energy remains constant. Energy continuously changes between kinetic energy and potential energy.
Spring potential energy
U = ½kx2
Kinetic energy
K = ½mv2
Total mechanical energy
E = K + U = ½mv2 + ½kx2 = ½kA2
At the extreme positions, speed is zero, so energy is entirely potential. At equilibrium, spring potential energy is zero for the ideal horizontal spring model, so energy is entirely kinetic.
7. Worked Numerical Example: Spring–Mass Oscillator
A 0.50 kg mass is attached to a spring with spring constant 200 N m−1. Find its angular frequency, period and frequency.
- Write the known values: m = 0.50 kg and k = 200 N m−1.
- Find angular frequency:
ω = √(k/m) = √(200/0.50) = √400 = 20 rad s−1 - Find the period:
T = 2π/ω = 2π/20 ≈ 0.314 s - Find the frequency:
f = 1/T ≈ 1/0.314 ≈ 3.18 Hz
Answer: The angular frequency is approximately 20 rad s−1, the period is 0.314 s, and the frequency is 3.18 Hz.
8. Formula and Symbol Guide
| Symbol | Meaning | Common SI unit |
|---|---|---|
| x | Displacement from equilibrium | m |
| A | Amplitude, or maximum displacement | m |
| t | Time | s |
| v | Instantaneous velocity | m s−1 |
| a | Instantaneous acceleration | m s−2 |
| m | Mass of the oscillating object | kg |
| k | Spring constant | N m−1 |
| T | Period | s |
| f | Frequency | Hz |
| ω | Angular frequency | rad s−1 |
| φ | Phase constant | rad |
| L | Length of a simple pendulum | m |
9. Common Mistakes and Misconceptions
- Forgetting the minus sign: In F = −kx and a = −ω2x, the minus sign shows the restoring direction.
- Confusing amplitude with total distance: Amplitude is the distance from equilibrium to one extreme. The distance between the two extremes is 2A.
- Mixing up period and frequency: Period is time per cycle; frequency is cycles per second. They are reciprocals.
- Using degrees in trigonometric calculations: SHM formulas use angular quantities in radians. Set a calculator to radian mode when required.
- Assuming every oscillation is SHM: Motion must have a restoring acceleration proportional to displacement and opposite in direction.
- Using the pendulum formula for large angles: The expression T = 2π√(L/g) is an approximation for small angular displacements.
- Thinking the object moves at constant speed: Speed changes throughout SHM: it is zero at the extremes and maximum at equilibrium.
10. Important Points to Remember
- SHM is periodic motion with a restoring force proportional to displacement and opposite in direction.
- The equilibrium position is the centre of the oscillation.
- For a spring, F = −kx and T = 2π√(m/k).
- For a small-angle simple pendulum, T = 2π√(L/g).
- Velocity is greatest at equilibrium; acceleration is greatest at the extreme positions.
- In ideal SHM, total mechanical energy remains constant while kinetic and potential energy exchange.
- Always use consistent SI units before substituting values into formulas.
Simple Harmonic Motion in One Minute
Simple harmonic motion is an oscillation in which the restoring acceleration is proportional to displacement and directed toward equilibrium.
a = −ω2x
For an ideal mass–spring system:
ω = √(k/m), T = 2π√(m/k), f = 1/T
For a simple pendulum at small angles:
T = 2π√(L/g)
At equilibrium, speed is maximum. At the extreme positions, speed is zero and the magnitude of acceleration is maximum. In ideal SHM, the total mechanical energy stays constant.