Composite and Cutout Moments of Inertia
The moment of inertia of a composite body is the signed sum of component inertias about the same final axis, with holes represented by negative mass and shifted using axis theorems when necessary.
Why this shows up in the exam
Perforated discs · Assemblies of rods and masses · Machined flywheels and plates
Learn the idea
Moments of inertia add for material pieces and subtract for removed pieces when all refer to one axis. Moments of inertia add for material pieces and subtract for removed pieces when all refer to one axis. 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: Add material, subtract holes, always about one axis.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- I_total = sum s_i I_i,same axis — Signed component sum with s_i = -1 for removed material.
- I_piece,final = I_piece,cm + M_piece d² — Shift each component to the common parallel axis when needed.
How to approach it
- 1Decompose into simple signed pieces
- 2Find each piece mass and centroidal inertia
- 3Shift every piece to the requested axis before summing
Common slip-ups that cost marks
- •Subtracting a hole with the full body's mass
- •Adding inertias about different axes
- •Ignoring density when finding removed mass
🌟 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.