Lesson 9 of 9 · 14 min

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Must-know facts

15 facts

1Every oscillatory motion is periodic; uniform circular motion is periodic but not oscillatory.
2ν = 1/T, ω = 2πν = 2π/T; 1 Hz = 1 s⁻¹. A heart at 75 beats per minute: 1.25 Hz, 0.8 s.
3SHM: x = A cos(ωt + φ); A amplitude, (ωt + φ) phase, φ phase constant.
4SHM is the projection of uniform circular motion on a diameter; radius = A, angular speed = ω.
5v = −ωA sin(ωt + φ), a = −ω²x; v_max = ωA at the mean position, a_max = ω²A at the extremes.
6Relative to x, the velocity differs in phase by π/2 and the acceleration by π.
7Speed at displacement x: v = ω√(A² − x²).
8F = −kx, k = mω², ω = √(k/m), T = 2π√(m/k).
9Block between two identical springs k on either side: T = 2π√(m/2k).
10E = ½kA² is constant; K and U each have period T/2.
11At x = A/2, U = E/4 and K = 3E/4.
12Simple pendulum, small angles: T = 2π√(L/g), independent of bob mass and amplitude.
13Seconds pendulum (T = 2 s): L ≈ 1 m for g = 9.8 m s⁻².
14sin² ωt has period π/ω; sin ωt − cos ωt = √2 sin(ωt − π/4) is SHM with period 2π/ω.
15The period of SHM does not depend on amplitude, energy or phase constant.

Common traps

Where marks are lost

Using ν and ω interchangeably, e.g. writing T = 1/ω.

ω = 2πν. T = 1/ν = 2π/ω. Check the units: ν in Hz (s⁻¹), ω in rad s⁻¹.

Putting degrees into x = A cos(ωt + φ) or into sin θ ≈ θ.

The phase and the small-angle rule use radians. 20° = 0.349 rad.

Saying the kinetic energy in SHM has the same period T as the displacement.

K and U depend on v² and x², which repeat twice per cycle, so their period is T/2 (frequency 2ν).

Thinking a heavier bob makes a pendulum swing more slowly.

Mass cancels from T = 2π√(L/g). Only length and g matter (for small swings).

Assuming that acceleration is zero where velocity is zero.

At the extremes v = 0 but a = ω²A, its largest value; at the mean position v is largest and a = 0.

Believing a larger amplitude gives a longer period in SHM.

T = 2π√(m/k) contains no A. A larger amplitude means a larger v_max and E, over the same time.

Treating every periodic motion, such as a bouncing ball or sin² ωt, as simple harmonic.

SHM needs a = −ω²x about a mean position. A bouncing ball has constant g between bounces; sin² ωt is harmonic only about ½.

Using k instead of 2k for a block held between two springs on either side.

Both springs push it back by kx each, so the effective constant is 2k and T = 2π√(m/2k).

Formulas

12 to know

Frequency and period

ν = 1/T

Unit hertz, 1 Hz = 1 s⁻¹.

Angular frequency

ω = 2π/T = 2πν

rad s⁻¹.

Displacement in SHM

x(t) = A cos(ωt + φ)

A amplitude, ωt + φ phase, φ phase constant.

Combining sine and cosine

A sin ωt + B cos ωt = D sin(ωt + φ), D = √(A² + B²), φ = tan⁻¹(B/A)

Same period 2π/ω.

Velocity in SHM

v(t) = −ωA sin(ωt + φ); v = ω√(A² − x²)

v_max = ωA at x = 0.

Acceleration in SHM

a(t) = −ω²A cos(ωt + φ) = −ω²x

a_max = ω²A at x = ±A.

Force law

F = −kx, k = mω²

Restoring force towards the mean position.

Spring-mass oscillator

ω = √(k/m), T = 2π√(m/k)

Two identical springs on either side: replace k by 2k.

Energies in SHM

K = ½k(A² − x²), U = ½kx², E = ½kA²

E constant; K and U have period T/2.

Pendulum (general, small angle)

T = 2π√(I/mgL)

I is the moment of inertia about the support.

Simple pendulum

T = 2π√(L/g)

Small angles; independent of bob mass.

Seconds pendulum length

L = gT²/4π²

T = 2 s, g = 9.8 m s⁻² gives L ≈ 1 m.

Key terms

16 terms

Periodic motion: Motion that repeats itself after equal intervals of time.
Oscillatory motion: Periodic back-and-forth motion about a mean (equilibrium) position.
Period: The least time after which a periodic motion repeats.
Frequency: Number of repetitions per second, the reciprocal of the period; unit hertz.
Displacement (in oscillations): The time-varying deviation of the oscillating quantity from its mean value: a position, an angle, a voltage, a pressure.
Simple harmonic motion: Oscillation in which displacement is a sinusoidal function of time, produced by a restoring force proportional to displacement.
Amplitude: The largest magnitude of displacement from the mean position.
Phase: The argument ωt + φ that fixes position and velocity at a given time.
Phase constant: The phase at t = 0.
Angular frequency: 2π times the frequency; ω in rad s⁻¹.
Reference circle: The circle whose uniformly moving point projects onto a diameter as SHM.
Restoring force: A force that always points back towards the equilibrium position.
Spring constant: The force per unit extension of a spring, k in F = −kx.
Linear harmonic oscillator: A system whose restoring force is exactly proportional to displacement.
Simple pendulum: A point bob on a massless, inextensible string swinging about a fixed support.
Seconds pendulum: A simple pendulum of period 2 s, about 1 m long on earth.

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