Moment of Inertia: Definition and Discrete Systems
The moment of inertia of particles about an axis is the sum of m_i r_perpendicular,i squared and quantifies resistance to angular acceleration about that exact axis.
Why this shows up in the exam
Point masses connected by light rods · Comparing mass distributions · Building inertia of particle assemblies
Learn the idea
Moment of inertia is the mass-weighted squared distance from a specified rotation axis. Moment of inertia is the mass-weighted squared distance from a specified rotation 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: Farther mass counts with distance squared.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- I_axis = sum m_i r_perp,i² — Moment of inertia of discrete masses about the named axis.
- I = integral r_perp² dm — Continuous-body definition.
How to approach it
- 1Mark the axis
- 2Find each perpendicular distance
- 3Sum m r squared and check units kg m squared
Common slip-ups that cost marks
- •Using distance from a point rather than the axis
- •Forgetting the square on distance
- •Treating moment of inertia as axis-independent
🌟 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.