Periodic and oscillatory motion
A motion that repeats itself after equal time gaps is periodic. The Earth going round the Sun and the hands of a clock are periodic.
If the object also moves to and fro about a middle point (the mean or equilibrium position), the motion is oscillatory. Every oscillation is periodic, but not every periodic motion is an oscillation (a clock hand goes round, not to and fro).
- Period T: time for one full oscillation. Unit: second (s).
- Frequency f: number of oscillations in one second. f = 1/T. Unit: hertz (Hz); 1 Hz = 1 oscillation per second.
- Angular frequency ω = 2πf = 2π/T. Unit: rad/s.
Displacement as a function of time
Displacement x is how far the object is from the middle, with a sign (+ on one side, − on the other). In SHM, x changes with time like a sine (or cosine) wave:
x(t) = A sin(ωt + φ)
- A = amplitude = the largest displacement.
- ω = angular frequency.
- φ = phase constant (starting phase): it fixes where the object was at t = 0.
A sine and a cosine are the same curve, just shifted, so x = A cos(ωt + φ′) is equally correct with a different φ′.
SHM, its equation and the reference circle
Simple harmonic motion is a to-and-fro motion in which the restoring force (the pull back to the middle) is proportional to the displacement and points opposite to it:
F = −kx so a = −ω²x, with ω = √(k/m).
The minus sign means the force always points back towards the middle. Differential form: d²x/dt² + ω²x = 0.
Reference circle: if a point moves round a circle of radius A at steady angular speed ω, its shadow on a diameter does SHM. This is why the same ω appears in circular motion and SHM.
Velocity and acceleration
- v = dx/dt = Aω cos(ωt + φ) = ±ω√(A² − x²). Largest (Aω) at the middle, zero at the ends.
- a = dv/dt = −Aω² sin(ωt + φ) = −ω²x. Largest (Aω²) at the ends, zero at the middle.
Phase and phase difference
The whole angle (ωt + φ) is called the phase. It tells the state of the motion: where the object is and which way it is moving.
- Phase 0: at the middle, moving to +x.
- Phase π/2: at +A, stopping for a moment.
- Phase π: back at the middle, moving to −x.
- Phase 3π/2: at −A.
Two SHMs with the same ω have a phase difference Δφ. If Δφ = 0 they move together (in phase); if Δφ = π they always move opposite (out of phase). Velocity leads displacement by π/2, and acceleration is π out of phase with displacement.
Energy in SHM
An oscillator swaps energy between two forms and keeps the total fixed (no friction):
- Kinetic energy K = ½mv² = ½k(A² − x²)
- Potential energy U = ½kx²
- Total energy E = K + U = ½kA² = ½mω²A² (constant)
At x = 0: K = E, U = 0. At x = ±A: K = 0, U = E. K = U when x = A/√2. K and U each repeat twice in one period, so they vary with period T/2.
Try it: a home pendulum clock
Tie a key or a small stone to a 1 m thread and let it swing a little. Use a phone stopwatch to time 20 full swings. Divide by 20 to get T. Then find f = 1/T. Now start it with a bigger swing (still small): T hardly changes. That is a sign of SHM. In the 3D above, move the A slider and check that T stays the same too.
Key formulas and definitions
- f = 1/T, ω = 2πf = 2π/T
- x = A sin(ωt + φ)
- F = −kx, a = −ω²x, ω = √(k/m)
- v = ω√(A² − x²), v_max = Aω, a_max = Aω²
- K = ½k(A² − x²), U = ½kx², E = ½kA² = ½mω²A²
Worked examples
1. A pendulum makes 40 oscillations in 20 s. Find its period, frequency and angular frequency.
T = 20/40 = 0.5 s. f = 1/T = 2 Hz. ω = 2πf = 4π ≈ 12.6 rad/s.
2. x = 0.05 sin(10πt + π/6) m. Find amplitude, frequency, period and initial phase.
Compare with x = A sin(ωt + φ): A = 0.05 m, ω = 10π rad/s, so f = ω/2π = 5 Hz, T = 0.2 s, φ = π/6.
3. A body in SHM has A = 4 cm and T = 2 s. Find the maximum speed and maximum acceleration.
ω = 2π/T = π rad/s. v_max = Aω = 0.04 × π ≈ 0.126 m/s. a_max = Aω² = 0.04 × π² ≈ 0.395 m/s².
4. For the body in the last example, find the speed when x = 2 cm.
v = ω√(A² − x²) = π√(0.04² − 0.02²) = π√(0.0012) ≈ π × 0.0346 ≈ 0.109 m/s.
5. A 0.2 kg mass does SHM with ω = 5 rad/s and A = 0.1 m. Find the total energy and the kinetic energy at x = 0.06 m.
k = mω² = 0.2 × 25 = 5 N/m. E = ½kA² = ½ × 5 × 0.01 = 0.025 J. U = ½ × 5 × 0.0036 = 0.009 J, so K = 0.025 − 0.009 = 0.016 J.
6. At what displacement is the kinetic energy equal to the potential energy?
K = U means ½k(A² − x²) = ½kx², so A² = 2x² and x = ±A/√2 ≈ ±0.71A.
7. Two SHMs: x₁ = 3 sin(ωt) and x₂ = 3 cos(ωt). Find the phase difference.
cos(ωt) = sin(ωt + π/2), so x₂ leads x₁ by π/2 (90°). When one is at the middle the other is at an end.
8. Which of these is SHM: (a) a = −9x, (b) a = −9x², (c) a = 9x?
Only (a): acceleration is proportional to x and opposite to it; ω = √9 = 3 rad/s, T = 2π/3 s. (b) is not proportional to x, (c) points away from the middle.
Common mistakes
- Thinking every periodic motion is SHM. A clock hand is periodic but not to and fro; SHM also needs F ∝ −x.
- Mixing f and ω. ω = 2πf, so a 2 Hz motion has ω ≈ 12.6 rad/s, not 2.
- Believing speed is highest at the ends. At the ends the body stops for a moment; speed is highest in the middle.
- Forgetting to convert cm to m before using E = ½kA². 5 cm = 0.05 m.