Kinetic Energy of Pure Rolling
A rigid body in pure rolling has total kinetic energy one-half M v_cm squared plus one-half I_cm omega squared, with omega = v_cm/R only when the no-slip condition holds.
Why this shows up in the exam
Comparing rolling rings and spheres · Stopping-work calculations · Inferring speed from rolling energy
Learn the idea
Rolling kinetic energy is the sum of centre translation and rotation about the centre. Rolling kinetic energy is the sum of centre translation and rotation about the centre. 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: Rolling energy has a glide part and a spin part.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- K = (1/2) M v_cm² + (1/2) I_cm omega² — Translation-plus-spin kinetic energy.
- K = (1/2)(M + I_cm/R²)v_cm² — Pure-rolling form after using omega=v_cm/R.
How to approach it
- 1Separate centre translation and spin
- 2Use I about the centre
- 3Apply the rolling constraint only if valid
Common slip-ups that cost marks
- •Counting translational energy only
- •Using inertia about the contact point while also adding translation
- •Imposing omega=v/R during slip
🌟 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.