Coupled Rolling Bodies, Strings, and Spools
For rolling bodies linked to strings, pulleys, planks, or other bodies, each component obeys its own translation and rotation equations while inextensible-string and no-slip constraints relate their accelerations.
Why this shows up in the exam
Rolling cylinders connected to hanging masses · Yo-yos and unwinding spools · Bodies rolling between moving surfaces
Learn the idea
Coupled translation and rotation require one consistent set of force, torque, string, and rolling constraints. Coupled translation and rotation require one consistent set of force, torque, string, and rolling constraints. 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: One equation per motion, then tie motions with constraints.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- sum F = M a_cm — Translation equation for each moving body.
- sum tau_cm = I_cm alpha — Rotation equation for each rotating body.
- a_string relations and a_cm = R alpha — Kinematic constraints imposed by strings and rolling contacts.
How to approach it
- 1Draw separate free-body diagrams
- 2Write translation and torque equations for each body
- 3Add string and rolling constraints with a declared sign convention
Common slip-ups that cost marks
- •Using one tension on a massive pulley
- •Writing incompatible string directions
- •Imposing no slip at a contact that is actually sliding
🌟 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.