Radius of Gyration
The radius of gyration k of a body about a specified axis is defined by I = M k squared, so k is the root-mean-square perpendicular distance of mass from that axis.
Why this shows up in the exam
Comparing compact and spread-out bodies · Rewriting pendulum and rolling formulas · Designing flywheel mass distributions
Learn the idea
Radius of gyration is the equivalent distance at which all mass could reproduce the same inertia. Radius of gyration is the equivalent distance at which all mass could reproduce the same inertia. 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: Radius of gyration is the RMS mass distance from the axis.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- k = sqrt(I/M) — Radius of gyration about the same axis used for I.
- k² = (1/M) integral r_perp² dm — Mean squared mass distance from the axis.
How to approach it
- 1Compute or identify I about the named axis
- 2Divide by total mass
- 3Take the positive square root and attach length units
Common slip-ups that cost marks
- •Treating k as the geometric radius
- •Using total mass inconsistent with I
- •Comparing k values about different axes
🌟 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.