SHM Graphs
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
In SHM, the x, v and a graphs are all sinusoids with the same period, shifted by quarter periods. Acceleration is always opposite to position (a = −ω²x), and velocity is zero exactly where x is at an extreme.
1. Lining up x, v and a
2. Reading graphs
Read the amplitude from the peak height and the period from peak to peak. Then ω = 2π/T gives vmax and amax.
From an a–t graph: A = amax/ω².
Direction questions: “moving in +x” means the x-graph slopes up (v > 0). “Acceleration in +x” means x is negative.
If x = A cos ωt, v = −Aω sin ωt starts at 0 and goes negative; a = −Aω² cos ωt starts at its most negative value.
3. Example, then try it
In the x–t graph below, find the oscillator’s maximum speed and maximum acceleration.
ω = 2π/2.0 = π rad/s.
vmax = (0.050)(π) ≈ 0.16 m/s; amax = (0.050)(π²) ≈ 0.49 m/s².
The a–t graph below is for an oscillator. Find ω, the amplitude and the maximum speed. (π/2 s ≈ 1.57 s)
ω: 2π ÷ (π/2) = 4.0 rad/s.
Amplitude: amax = Aω² → A = 8.0 ÷ 16 = 0.50 m.
Max speed: vmax = Aω = (0.50)(4.0) = 2.0 m/s.
Answer: ω = 4.0 rad/s, A = 0.50 m, vmax = 2.0 m/s.
4. Common mistakes
At the instant an oscillator is at its maximum positive displacement, what are its velocity and acceleration?
At x = +A the object turns around (v = 0), and a = −ω²x is at its most negative.
Ready to practice?
5 questions on SHM Graphs, each with a hint and an explanation for every wrong answer.