Standard Moments of Inertia
A standard moment-of-inertia formula is valid only for its stated ideal body and axis, such as a ring, disc, cylinder, sphere, rod, or rectangular lamina of uniform density.
Why this shows up in the exam
Selecting inertia for rotational dynamics · Comparing rings, discs, cylinders, and spheres · Constructing composite-body inertia
Learn the idea
Symmetric bodies have standard axis-specific inertia formulas that encode their mass distribution. Symmetric bodies have standard axis-specific inertia formulas that encode their mass distribution. 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: Formula, body, and axis must travel together.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- I_ring,axis = M R² — Thin ring about its central symmetry axis.
- I_disc,axis = (1/2) M R² — Uniform thin disc about its central symmetry axis.
- I_solid sphere,diameter = (2/5) M R² — Uniform solid sphere about a diameter.
- I_rod,centre = (1/12) M L² — Thin uniform rod about a central perpendicular axis.
How to approach it
- 1Sketch the idealized body and axis
- 2Select the matching standard result
- 3Use an axis theorem only after the centroidal formula is correct
Common slip-ups that cost marks
- •Remembering a coefficient but not its axis
- •Using thin-body formulas for thick bodies
- •Confusing solid and hollow objects
🌟 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.