Moment of Inertia from Nonuniform Density
For a body whose line, area, or volume density varies with position, moment of inertia is obtained from I = integral r_perpendicular squared dm using the stated density law and geometric Jacobian.
Why this shows up in the exam
Discs with radial density gradients · Rods with position-dependent linear density · Custom flywheel profiles
Learn the idea
Nonuniform bodies require integrating squared axis distance with the actual spatial mass density. Nonuniform bodies require integrating squared axis distance with the actual spatial mass density. 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: Density tells how much mass each distance-squared contribution carries.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- I = integral r_perp² dm — General inertia integral about the selected axis.
- dm = lambda(x) dx or sigma(r) dA or rho dV — Mass element carrying the nonuniform density.
How to approach it
- 1Write density and the correct differential element
- 2Set limits over actual material
- 3Integrate mass and inertia consistently before forming ratios
Common slip-ups that cost marks
- •Normalizing density incorrectly
- •Using r from the origin instead of distance to the axis
- •Dropping the polar area factor r dr dtheta
🌟 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.