C.1 · IB Physics
SHM: displacement velocity and acceleration
Watch a mass on a spring move in step with a rotating reference circle, and compare the x, v and a graphs to see the phase differences of π/2 and π. Change the phase constant φ and see the whole motion shift.
ω = 2π / T
x = A sin(ωt + φ)
v = Aω cos(ωt + φ)
a = −Aω² sin(ωt + φ) = −ω²x
x = A sin(ωt + φ)
v = Aω cos(ωt + φ)
a = −Aω² sin(ωt + φ) = −ω²x
t = 0
ωt + φ = 0
x = 0
v = 0
a = 0
ω = 0
v leads x by π/2; a and x differ by π.
Try this
- Pause when the mass is at its highest point. Check that v = 0 and a is at its most negative, and that the a phasor points exactly opposite the x phasor.
- Note the shape of the v graph, then set φ = π/2. Compare the new x graph with the old v graph: what does this tell you about the phase of v?
- Keep A fixed and double T from 1.00 s to 2.00 s. By what factor do the peak velocity and peak acceleration change?
Exam tip: to read a phase difference from two graphs, measure the time shift between matching peaks and use Δφ = 2π × (Δt / T); a quarter-period shift is π/2 and a half-period shift is π.