Simple harmonic motion flashcards

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Simple harmonic motion
Oscillation in which the acceleration is proportional to the displacement from a fixed equilibrium point and always directed towards that point.
Equilibrium position (of an oscillator)
The position where the resultant force on the oscillating object is zero; displacement is measured from it.
Restoring force
A force that acts to return a displaced object towards its equilibrium position. For SHM it is proportional to the displacement.
Amplitude
The maximum displacement from the equilibrium position.
Period T
The time taken for one complete oscillation. Unit s.
Angular frequency ω
2π × frequency (rad s⁻¹); the constant in a = −ω²x.
Simple pendulum
A small, dense bob on a light, inextensible string; length l is measured from the pivot to the centre of the bob.
Condition for SHM (force and acceleration forms)
F = −kx and a = −ω²x (Yes)
Displacement, velocity and acceleration when timing starts at x = +A
x = A cos ωt; v = −Aω sin ωt; a = −Aω² cos ωt (ωt in radians) (Yes)
Links between T, f and ω
T = 1/f = 2π/ω; ω = 2πf (Yes)
Period of a mass on a spring
T = 2π√(m/k) (Yes)
Period of a simple pendulum
T = 2π√(l/g), small angles only (Yes)
Maximum speed and maximum acceleration in SHM
v_max = Aω (at equilibrium); a_max = Aω² (at the extremes) (No — derive it)
Straight-line form of the pendulum equation
T² = (4π²/g) l: plot T² against l, gradient 4π²/g. (No — derive it)
Shape of the a–x graph for SHM
A straight line through the origin with negative gradient −ω².
Gradient of a displacement–time graph for SHM
The velocity. Zero at the peaks and troughs; steepest where the graph crosses the time axis.
Gradient of a velocity–time graph for SHM
The acceleration. Steepest where v = 0 (the extremes), zero at the peaks of v (equilibrium).
Phase of velocity relative to displacement in SHM
Velocity leads displacement by a quarter of a cycle (π/2 rad).
Phase of acceleration relative to displacement in SHM
In antiphase: π rad (180°) apart. When x is most positive, a is most negative.
x–t graph when timing starts as the object passes equilibrium
A sine curve, x = A sin ωt, instead of the cosine curve.
Why is a ball bouncing on a hard floor not SHM?
Between bounces its acceleration is g downwards at every height — not proportional to displacement.
Why is a mass on a spring SHM?
By Hooke's law the resultant force is −kx about the equilibrium position, so a = −(k/m)x with ω² = k/m.
Why does the pendulum formula need small angles?
The restoring force is mg sin θ; only for small θ (in rad) is sin θ ≈ θ = x/l, making the force proportional to x.
Why does a pendulum's period not depend on the bob's mass?
Mass appears in both the restoring force (mg sin θ) and the inertia (a = F/m), so it cancels.
Does a mass–spring system have a different period on the Moon?
No. g only moves the equilibrium position; T = 2π√(m/k) has no g in it.
Where in SHM are the speed and the acceleration largest?
Speed largest at equilibrium; acceleration largest at the extremes (±A), where the speed is zero.
Why time 20 oscillations rather than one?
The reaction-time uncertainty is shared over 20 periods, so the percentage uncertainty in T is 20 times smaller.
SHM with A = 0.050 m and f = 2.0 Hz. Maximum speed?
ω = 2π × 2.0 = 12.6 rad s⁻¹; v_max = 0.050 × 12.57 = 0.63 m s⁻¹.
Period of a 0.20 kg mass on a spring with k = 45 N m⁻¹?
T = 2π√(0.20/45) = 0.42 s. (Yes)
Length of a simple pendulum with period 1.5 s?
l = gT²/4π² = 9.81 × 1.5² / 4π² = 0.56 m. (Yes)
T = 0.40 s. Acceleration magnitude at x = 0.012 m?
ω = 2π/0.40 = 15.7 rad s⁻¹; |a| = ω²x = 15.7² × 0.012 = 3.0 m s⁻².
x = 0.20 cos(3.0t) m. Displacement at t = 0.50 s?
ωt = 1.5 rad; x = 0.20 cos 1.5 = 0.014 m (calculator in radians).
A coin rests on a plate vibrating vertically at 8.0 Hz. Largest amplitude before it leaves the plate?
Aω² = g → A = 9.81/(2π × 8.0)² = 3.9 × 10⁻³ m (3.9 mm).
Calculator mode for x = A cos ωt
Radians. In degree mode cos ωt stays near 1 and the object seems not to move.
Amplitude of a hanging mass pulled down and released
The extra distance pulled down from equilibrium — not the static extension of the spring.
Testing data for SHM
Show that a/x is the same negative constant for several readings, or that a–x is a straight line through the origin.
Percentage uncertainty in T²
Double the percentage uncertainty in T.
Where to start and stop the stopwatch for a pendulum
As the bob passes the equilibrium position (fiducial mark), where it moves fastest.

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