Short on time? This page is the recap for the calculus-based course (separate from Physics 1, which uses no calculus). For explanations, graphs and worked examples, use the full study guide.
Recap
- Measure x from equilibrium (net force zero) and state the positive direction.
- Newton’s second law with a restoring force F_x = −kx gives d²x/dt² = −ω²x, ω = √(k/m). Any system that reduces to this form is SHM.
- The solution is x = A cos(ωt + φ₀). You must know it and use it; you do not have to prove it.
- A and φ₀ come from the start (x₀ and v_x0). ω comes from the system.
- Differentiate for v_x and a_x. Read zeros and extremes from the table below.
- Resonance: a sinusoidal driving force at the natural frequency adds energy every cycle, so the amplitude grows.
Key relationships
| Relationship | Units | Notes |
|---|---|---|
| d²x/dt² = −ω²x | m/s² | from Newton’s second law; ω² is the constant |
| x = A cos(ωt + φ₀) | m | sine form: φ₁ = φ₀ + π/2 |
| v_x = −Aω sin(ωt + φ₀) | m/s | slope of x–t graph |
| a_x = −Aω² cos(ωt + φ₀) = −ω²x | m/s² | a_x–x graph: straight line, slope −ω² |
| v_max = Aω; a_max = Aω² | m/s; m/s² | at x = 0 and at x = ±A |
| |v_x| = ω√(A² − x²) | m/s | speed at any position |
| A = √(x₀² + (v_x0/ω)²); cos φ₀ = x₀/A, sin φ₀ = −v_x0/(Aω) | m; rad | use both signs for the quadrant |
| T = 2π/ω = 1/f | s | no A in it |
Assumptions
- Restoring force exactly proportional to displacement; no friction.
- Angles in radians (ω in rad/s).
- Equilibrium is the origin, so x(t) has no constant offset.
Mistakes to avoid
- Fastest at the ends. The object is at rest at x = ±A.
- a_x = 0 so v_x = 0. At equilibrium a_x = 0 and the speed is greatest.
- Bigger A, longer T. T does not depend on A in SHM.
- v_max = Af. It is Aω, with ω = 2πf.
- Degree mode. ωt + φ₀ is in radians.
- φ₀ from tan alone. Check the signs of cos φ₀ and sin φ₀.
- Any periodic motion is SHM. Only if the restoring force is proportional to displacement.
Quick self-check
- x = (0.20 m) cos(5.0t). What are v_max and a_max? (Aω = 1.0 m/s; Aω² = 5.0 m/s²)
- A cart starts at x₀ = 0 with v_x0 = −0.60 m/s, and ω = 4.0 rad/s. What is A, and which form of x(t) is simplest? (A = 0.15 m; x = −(0.15 m) sin(4.0t))
- An a_x–x graph is a straight line through the origin with slope −49 s⁻². What is the period? (ω = 7.0 rad/s, T = 2π/7.0 = 0.90 s)
Next: practice questions.