Circular Motion and Gravitation
This section covers Circular Motion and Gravitation. The exam tests angular quantities, centripetal acceleration and force, gravitational fields, and satellite orbits.
Core rules
- Angular velocity ω = 2π/T = v/r. Centripetal acceleration is a = v²/r = ω²r and is directed towards the centre of the circle.
- Centripetal force is the resultant inward force, F = mv²/r. It is not a new kind of force; tension, gravity or friction may supply it.
- Newton's law of gravitation gives F = GMm/r². Gravitational field strength g = F/m = GM/r² outside a uniform sphere.
- For a satellite in a circular orbit, gravity provides the centripetal force: GMm/r² = mv²/r, so v = √(GM/r). The satellite's mass cancels.
- Orbital period satisfies T² ∝ r³ for orbits around the same central body.
Worked example
A 1200 kg car travels at 15 m/s around a circular bend of radius 50 m. Its centripetal acceleration is a = v²/r = 15²/50 = 225/50 = 4.5 m s⁻². The required centripetal force is F = ma = 1200 × 4.5 = 5400 N, directed towards the centre of the bend. This force is provided by friction between the tyres and the road.
Traps
Do not add an outward centrifugal force in an inertial frame; the resultant force is inward. Speed can be constant while velocity changes because direction changes. A satellite's mass cancels from orbital speed but affects the gravitational force on it. Gravitational field strength is not the same as gravitational potential. Inside a uniform sphere, g depends on distance from the centre and is not GM/r².