Lesson 9 of 9 · 14 min
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Must-know facts
15 facts
- 01Every oscillatory motion is periodic; uniform circular motion is periodic but not oscillatory.
- 02ν = 1/T, ω = 2πν = 2π/T; 1 Hz = 1 s⁻¹. A heart at 75 beats per minute: 1.25 Hz, 0.8 s.
- 03SHM: x = A cos(ωt + φ); A amplitude, (ωt + φ) phase, φ phase constant.
- 04SHM is the projection of uniform circular motion on a diameter; radius = A, angular speed = ω.
- 05v = −ωA sin(ωt + φ), a = −ω²x; v_max = ωA at the mean position, a_max = ω²A at the extremes.
- 06Relative to x, the velocity differs in phase by π/2 and the acceleration by π.
- 07Speed at displacement x: v = ω√(A² − x²).
- 08F = −kx, k = mω², ω = √(k/m), T = 2π√(m/k).
- 09Block 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
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.