Velocities of Points on a Rolling Body
For a rigid body in planar motion, the velocity of point P equals v_cm plus omega cross r from the centre of mass to P; in pure rolling the contact point is instantaneously at rest and the top point has speed 2v_cm.
Why this shows up in the exam
Finding rim-point speeds · Locating instantaneous centres · Analysing nested rolling discs
Learn the idea
Each point's ground velocity is the centre velocity plus its velocity relative to the centre. Each point's ground velocity is the centre velocity plus its velocity relative to 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: Add the centre's glide to the point's spin velocity.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- v_P = v_cm + omega cross r_P/CM — Rigid-body point-velocity relation.
- v_contact = 0, v_top = 2 v_cm — Special speeds for pure rolling on a stationary plane.
How to approach it
- 1Draw v_cm and the local tangential velocity
- 2Add them vectorially
- 3Check contact and top-point limiting cases
Common slip-ups that cost marks
- •Assigning every point speed v_cm
- •Using the instantaneous centre for acceleration as if fixed
- •Ignoring vector directions
🌟 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.