Short on time? This page is the recap. For the derivations, figures and three worked examples, use the full study guide.
Recap
- On a circle the velocity’s direction changes, so the object accelerates even at constant speed.
- Differentiating r(t) = r cos(ωt) î + r sin(ωt) ĵ twice gives a = −ω²r: size v²/r, pointing to the centre.
- If the speed changes, add tangential acceleration a_t = dv/dt, perpendicular to a_c.
- Take + toward the centre for the radial equation. “Centripetal force” is the inward net force, never an extra arrow.
Key relationships
| Relationship | Condition or meaning |
|---|---|
| a_c = v²/r = 4π²r/T² | toward the centre; v²/r holds on any circular path, the T form only at constant speed |
| a_t = dv/dt = d²s/dt² | along the velocity; zero for uniform circular motion |
| |a| = √(a_c² + a_t²) | net acceleration |
| T = 1/f; T = 2πr/v | uniform circular motion |
| ΣF toward centre = mv²/r | Newton’s second law along the radius |
| v_min = √(gr) | top of a vertical loop, when N (or tension) is zero |
| v² = rg tan θ | banked curve needing no friction |
| v_max² = rg(sin θ + μ_s cos θ)/(cos θ − μ_s sin θ) | banked curve, friction down the slope |
| tan θ = v²/(rg); T = 2π√(L cos θ/g) | conical pendulum |
| T² = (4π²/GM) r³ | circular orbit, gravity the only force |
For v_min on a banked curve, change the sign of every μ_s term. If tan θ ≤ μ_s there is no minimum.
Assumptions
- Inertial reference frame; objects treated as points.
- g = 9.8 m/s² near Earth’s surface; orbit radius measured from the planet’s centre.
- Kepler’s first and second laws are not needed: orbits here are circles.
Mistakes to avoid
- Drawing a “centripetal force” arrow on a free-body diagram.
- Assuming the net force points to the centre when the speed is changing.
- Fixing friction’s direction on a banked curve before checking the speed.
- Using altitude instead of orbit radius.
- Normal force up at the top of a loop (inside the track, it points down).
- Using frequency in place of 2πf when finding speed: v = 2πrf.
Quick self-check
- A car goes round a bend of radius 16 m at 8.0 m/s. Find a_c. (64 ÷ 16 = 4.0 m/s²)
- A ball on a string moves in a circle of radius 0.50 m at 2.0 Hz. Find T, v and a_c. (T = 0.50 s; v = 2π × 0.50 × 2.0 ≈ 6.3 m/s; a_c ≈ 79 m/s²)
- What is the minimum speed at the top of a loop of radius 0.90 m? (√(9.8 × 0.90) ≈ 3.0 m/s)
Next: practice questions.