Short on time? This page is the recap for the calculus-based course (separate from Physics 1). For explanations, figures and worked examples, use the full study guide.
Recap
- Periodic motion repeats in equal time intervals. SHM is a special case of it.
- The equilibrium position is where the net force is zero. Measure displacement x from there.
- A restoring force points opposite to the displacement, back toward equilibrium.
- SHM happens when the restoring (net) force is proportional to the displacement: m a_x = −kΔx, with Δx measured from equilibrium (F_net = −kx when x = 0 is at equilibrium).
- A constant extra force (such as gravity on a vertical spring) only shifts the equilibrium; the motion is still SHM with the same k.
- Near a stable equilibrium of any smooth U(x), small oscillations are approximately SHM.
Key relationships
| Relationship | Meaning |
|---|---|
| m a_x = −kΔx | SHM condition: Newton’s second law with a restoring force (Δx from equilibrium, k > 0 and constant) |
| F_net,x = −kx | the same condition with the origin at equilibrium |
| d²x/dt² = −(k/m)x | Newton’s second law for SHM; any variable obeying this form is in SHM |
| F₀ − kx = −k(x − F₀/k) | constant force F₀ shifts equilibrium to x_e = F₀/k |
| d = mg/k | static stretch of a vertical spring; net force about equilibrium is −ky |
| F_x = −dU/dx; k_eff = d²U/dx² at a minimum | small oscillations about a stable equilibrium |
| d²θ/dt² = −(g/ℓ) sin θ ≈ −(g/ℓ)θ | simple pendulum, small angles in radians |
Assumptions behind the numbers
- Ideal (massless, linear) springs; smooth surfaces; no air resistance unless stated.
- g = 9.8 m/s². Angles in radians for sin θ ≈ θ.
- “Small oscillation” means displacements small enough that the linear force model is accurate.
Mistakes to avoid
- Calling any periodic motion SHM. Test the force law.
- Using the spring force as the restoring force when gravity also acts. Use the net force.
- Measuring x from the relaxed spring length when a constant force shifts equilibrium.
- Missing the sign. F = +kx is not restoring; the equilibrium is unstable.
- Applying k_eff = U″ at a maximum. U″ < 0 there: no oscillation.
- Using degrees in sin θ ≈ θ.
Quick self-check
- A 0.25 kg object has F_net = −(50 N/m)x. What is its acceleration at x = 0.040 m? (−8.0 m/s²)
- A 0.50 kg block hangs from a spring with k = 98 N/m. How far is the spring stretched at equilibrium? (0.050 m)
- U(x) = (5.0 J/m²)(x − 0.20 m)². Is this SHM? About where, and with what k? (Yes, about x = 0.20 m, k = d²U/dx² = 10 N/m)
Next: practice questions.