Short on time? This page is the recap. For diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Velocity is always tangent to the circle. On a circle its direction changes, so the object accelerates even at constant speed.
- Centripetal acceleration is the part of the acceleration towards the centre. Tangential acceleration is the part along the path; it changes the speed.
- The inward force is not a new force. Tension, friction, normal force, gravity, or components of them, add up to m v²/r towards the centre.
- At the top of a vertical loop at the minimum speed, gravity alone supplies the inward force.
- On a frictionless banked curve, N sin θ supplies the inward force. In a conical pendulum, F_T sin θ does.
- In a circular orbit, gravity alone supplies the inward force, and the satellite’s mass cancels.
Key relationships
| Relationship | Use it to |
|---|---|
| a_c = v² / r (towards centre) | find the inward acceleration from speed and radius |
| T = 1 / f | switch between period (s) and frequency (Hz) |
| v = 2πr / T, so a_c = 4π²r / T² | link speed, radius and period at constant speed |
| ΣF_in = m v² / r | apply Newton’s second law along the inward axis |
| a = √(a_c² + a_t²) | find the net acceleration when the speed changes |
| v_min = √(g r) at the top of a loop | find the slowest speed that keeps contact |
| tan θ = v² / (r g) | frictionless banked curve; also a conical pendulum (θ from vertical) |
| T² = (4π² / GM) r³ | circular orbit around a central mass M (Kepler’s third law) |
Assumptions behind the numbers
- Objects are treated as points; strings are light and do not stretch.
- Banked-curve calculations assume no friction. With friction you only describe its direction.
- Orbit radius r is measured from the centre of the central body, and the satellite is much lighter than it.
- g = 9.8 m/s²; G = 6.67 × 10⁻¹¹ N·m²/kg².
Mistakes to avoid
- Adding an F_c arrow to a free-body diagram.
- Thinking constant speed means no acceleration.
- Using string length L as the radius of a conical pendulum. Use r = L sin θ.
- Assuming N = mg at the top or bottom of a loop or on a bank.
- Saying a released object flies outward. It leaves along the tangent.
- Mixing up factor-of-change cases. Doubling r at the same speed halves a_c; doubling r at the same period doubles a_c.
- Measuring orbit radius from the planet’s surface.
Quick self-check
- v = 6.0 m/s on a circle of radius 3.0 m. What is a_c? (12 m/s² towards the centre)
- A wheel turns at 2.5 Hz. What is its period? (0.40 s)
- What is the minimum speed at the top of a vertical loop of radius 2.5 m? (√(9.8 × 2.5) = 4.9 m/s)
- a_c = 3.0 m/s² and a_t = 4.0 m/s². What is the size of the net acceleration? (5.0 m/s²)
Next: practice questions.