Simple Harmonic Motion
Simple Harmonic Motion
Simple harmonic motion (SHM) is a special type of oscillatory motion in which the acceleration is always directed toward a fixed equilibrium position and is proportional to the displacement from that position.
Defining Equation
a = −ω²x
where:
- a = acceleration (m s⁻²)
- ω = angular frequency (rad s⁻¹)
- x = displacement from equilibrium (m)
- The negative sign indicates the acceleration is always directed opposite to the displacement (toward equilibrium)
This is the defining condition for SHM. If a system satisfies this equation, it undergoes SHM.
Key Equations of SHM
Displacement: x = A cos(ωt) or x = A sin(ωt)
(The choice depends on initial conditions: cos if timing starts at maximum displacement, sin if starting at equilibrium.)
Velocity: v = −Aω sin(ωt) or v = Aω cos(ωt)
Maximum velocity: v_max = Aω (at equilibrium, x = 0)
Velocity as a function of displacement: v = ±ω√(A² − x²)
Acceleration: a = −Aω² cos(ωt) = −ω²x
Maximum acceleration: a_max = Aω² (at maximum displacement, x = ±A)
Period and Frequency
ω = 2πf = 2π/T
The period T is independent of amplitude — this is a key characteristic of SHM called isochronous oscillation.
Energy in SHM
Kinetic energy: E_k = ½mv² = ½mω²(A² − x²)
Potential energy: E_p = ½mω²x² (= ½kx² for a spring)
Total energy: E_total = ½mω²A² = constant
At equilibrium (x = 0): all energy is kinetic (E_k = max, E_p = 0)
At maximum displacement (x = ±A): all energy is potential (E_k = 0, E_p = max)
Energy is continuously exchanged between kinetic and potential forms, but the total mechanical energy remains constant (in the absence of damping).
The Mass-Spring System
A mass m on a spring of spring constant k:
Restoring force: F = −kx → ma = −kx → a = −(k/m)x
Comparing with a = −ω²x: ω² = k/m
Period: T = 2π√(m/k)
Derivation
From Newton's second law: m(d²x/dt²) = −kx
This gives d²x/dt² = −(k/m)x, which has the solution x = A cos(ωt) with ω = √(k/m).
Therefore T = 2π/ω = 2π√(m/k)
The Simple Pendulum
For a pendulum of length L making small angle oscillations (θ < ~10°):
The restoring force along the arc is F = −mg sin θ ≈ −mgθ (small angle approximation)
Since x = Lθ: F = −mg(x/L) → a = −(g/L)x
So ω² = g/L and T = 2π√(L/g)
Note: The period depends only on L and g, not on the mass or the amplitude (for small oscillations).
Worked Example
A 0.40 kg mass oscillates on a spring (k = 25 N m⁻¹) with amplitude 0.060 m. Find:
(a) Period:
T = 2π√(m/k) = 2π√(0.40/25) = 2π√(0.016) = 2π × 0.1265 = 0.795 s
(b) Maximum speed:
ω = 2π/T = 2π/0.795 = 7.90 rad s⁻¹
v_max = Aω = 0.060 × 7.90 = 0.474 m s⁻¹
(c) Maximum acceleration:
a_max = Aω² = 0.060 × 7.90² = 0.060 × 62.4 = 3.75 m s⁻²
(d) Speed when x = 0.030 m:
v = ω√(A² − x²) = 7.90 × √(0.060² − 0.030²) = 7.90 × √(0.0027) = 7.90 × 0.05196 = 0.410 m s⁻¹
Graphical Representation
The displacement, velocity, and acceleration graphs are all sinusoidal:
- Velocity leads displacement by π/2 (90°) — v is maximum when x = 0
- Acceleration leads velocity by π/2 and is in antiphase with displacement
- The a-x graph is a straight line through the origin with gradient −ω² (this is the test for SHM)
Damped Oscillations
In real systems, energy is lost to the surroundings (friction, air resistance), and the amplitude decreases over time.
- Light damping: Amplitude decreases exponentially; the system oscillates many times before stopping; period approximately unchanged
- Heavy damping: Few oscillations before stopping; period slightly longer
- Critical damping: Returns to equilibrium in the minimum time without oscillating (e.g., car shock absorbers)
- Overdamping: Returns to equilibrium without oscillating but takes longer than critical damping (e.g., fire doors)
Forced Oscillations and Resonance
When a periodic driving force acts on an oscillating system:
- If driving frequency << natural frequency: amplitude is small, oscillation in phase with driver
- If driving frequency = natural frequency (f₀): resonance — maximum amplitude, phase lag = π/2
- If driving frequency >> natural frequency: amplitude is small, oscillation in antiphase with driver
Required Practical: Investigating SHM of a Mass-Spring System
1. Attach masses to a spring and measure T for ~10 oscillations
2. Plot T² against m — should be a straight line through the origin
3. Gradient = 4π²/k → determine k
4. Verify T is independent of amplitude by varying A and measuring T