Central Forces and Orbital Angular Momentum
When force is always parallel or antiparallel to the radius vector from a fixed centre, its torque about that centre is zero, so particle angular momentum and the rate at which radius sweeps area remain constant.
Why this shows up in the exam
Planetary and satellite motion · Particles in inverse-power central forces · Relating orbital radius, speed, and period
Learn the idea
A central force exerts zero torque about its centre, conserving orbital angular momentum and areal velocity. A central force exerts zero torque about its centre, conserving orbital angular momentum and areal velocity. Start from a clear axis, origin, body, and reference frame; the geometry and constraints then decide which rotational law is safe to use.
🧠 Memory hook: Radial force cannot twist about the centre.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- tau_centre = r cross F(r) = 0 — Zero torque for a central force.
- L = m r² theta_dot = constant — Orbital angular momentum in plane polar coordinates.
- dA/dt = L/(2m) — Constant areal velocity under a central force.
How to approach it
- 1Identify the force centre
- 2Test r cross F about that centre
- 3Apply angular-momentum conservation and the radial force equation separately
Common slip-ups that cost marks
- •Calling every circular force central without checking its line
- •Conserving speed when only angular momentum is conserved
- •Using distance from a different origin
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
Masses 1 kg and 3 kg lie at x = 0 and x = 4 m. Find the x-coordinate of their centre of mass.
More from Motion of System of Particles and Rigid Body
Conservation of momentum and angular momentum
The total linear and angular momentum of a system remains constant in the absence of external forces or torques, including during collisions and rotational motion.
Moment of inertia and radius of gyration
Moment of inertia quantifies how mass is distributed with respect to an axis of rotation, and the radius of gyration is a measure related to this distribution.
Torque and rotational equilibrium
Torque is the rotational analogue of force, causing angular acceleration, and equilibrium occurs when the net torque on a body is zero.
Center of mass: definition and calculation
The center of mass is the point representing the mean position of the mass in a system, and can be calculated for discrete particles or continuous bodies.
Rotational kinematics and dynamics
Rotational kinematics describes the motion of rotating bodies, while dynamics relates torque, angular acceleration, and rotational kinetic energy.
Centre of Mass of Discrete Particles
For discrete particles, the centre-of-mass position is the vector sum of each mass times its position divided by total mass; this point governs translation even when the particles move relative to one another.