Short on time? This page is the recap. For explanations, diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Periodic motion repeats itself in equal time intervals. SHM is a special case of it.
- The equilibrium position is where the net force on the object is zero.
- A restoring force points opposite to the displacement from equilibrium, whichever way the object is moving.
- SHM condition: the size of the restoring force is proportional to the size of the displacement. A graph of F_net (or a) against x is a straight line through the origin with a negative slope.
- A vertical spring is still SHM. Gravity just moves the equilibrium down by mg/k.
- A pendulum has a restoring torque mgℓ sin θ. For small angles (in radians) sin θ ≈ θ, so the torque is proportional to θ and the motion is close to SHM.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| F_net = −kx | k in N/m, x in m from equilibrium | test for SHM; find the restoring force |
| a = −(k/m)x | a in m/s² | find acceleration at any displacement |
| d = mg / k | d in m | find where a vertical spring’s equilibrium is |
| τ = mgℓ sin θ ≈ mgℓθ | θ in rad, τ in N·m | restoring torque on a pendulum (small angles) |
| F ≈ (mg/ℓ)s along the arc | s = ℓθ in m | treat a small-angle pendulum like a spring |
Assumptions behind the numbers
- Springs and strings are ideal: massless, and the spring obeys Hooke’s law.
- No friction or air resistance, so nothing but the restoring force changes the motion along the axis.
- A pendulum bob is a point mass on a light string, swinging through a small angle (about 15° or less keeps the error near 1%).
- g = 9.8 m/s².
Mistakes to avoid
- Calling all periodic motion SHM. A bouncing ball repeats, but its force is not −kx.
- Measuring x from the wrong place. Use the equilibrium position, not the relaxed length of a hanging spring.
- Giving the restoring force the direction of motion. It points toward equilibrium.
- “Acceleration is biggest at equilibrium.” It is zero there.
- Using degrees in sin θ ≈ θ. Radians only.
- Forgetting the sign. a and x always have opposite signs in SHM.
Quick self-check
- A 0.50 kg block on a spring with k = 200 N/m is at x = +0.10 m from equilibrium. What is its acceleration? (−40 m/s², toward equilibrium)
- A 0.30 kg mass hangs at rest from a spring with k = 60 N/m. How far is the spring stretched? (0.049 m)
- An object in SHM has an acceleration of size 1.5 m/s² at x = +2.0 cm. What is its acceleration at x = −6.0 cm? (+4.5 m/s²: three times as far, opposite side)
- Is a puck sliding back and forth between two walls in SHM? (No: the force is zero except at the walls, so it is not proportional to x.)
Next: practice questions.