Pure Rolling Constraint
For a rigid circular body rolling without slipping on a stationary surface, the centre-of-mass speed and angular speed satisfy v_cm = R omega, with corresponding tangential accelerations when the constraint remains valid.
Why this shows up in the exam
Wheels rolling on roads · Cylinders descending rough inclines · Relating distance travelled to angular displacement
Learn the idea
Pure rolling combines translation and rotation with zero relative velocity at the contact point. Pure rolling combines translation and rotation with zero relative velocity at the contact point. 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: Pure rolling means the contact point is instantaneously at rest.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- v_cm = R omega — No-slip velocity constraint on a stationary surface.
- a_cm,tangent = R alpha — Tangential acceleration constraint while pure rolling persists.
How to approach it
- 1State the surface frame
- 2Check whether no slip is given or dynamically possible
- 3Apply signed translational and angular constraints
Common slip-ups that cost marks
- •Assuming zero acceleration at the contact point
- •Using the constraint during slipping
- •Ignoring the velocity of a moving surface
🌟 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
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Moment of inertia and radius of gyration
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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.