Transition from Sliding to Rolling
While a body slips on a rough surface, kinetic friction produces linear acceleration and angular acceleration independently; pure rolling begins at the first instant v_cm = R omega in the surface frame.
Why this shows up in the exam
Discs projected without spin · Spinning balls dropped on rough floors · Finding energy lost before rolling begins
Learn the idea
Kinetic friction changes translation and spin until the contact-point relative velocity becomes zero. Kinetic friction changes translation and spin until the contact-point relative velocity becomes zero. 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: Friction drives the contact slip speed toward zero.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- M dv_cm/dt = F_ext — Translational evolution during slip.
- I_cm domega/dt = tau_cm — Spin evolution during slip.
- v_cm - R omega = 0 — Condition marking the onset of pure rolling for the chosen signs.
How to approach it
- 1Write the signed contact relative velocity
- 2Use kinetic friction for v and omega evolution
- 3Solve for its first zero and then switch to rolling equations
Common slip-ups that cost marks
- •Using static friction during gross slip
- •Assuming mechanical energy is conserved
- •Stopping the calculation when v or omega separately reaches zero
🌟 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.