Question
A uniform solid sphere of mass and radius rolls without slipping down a smooth incline of angle . Find the linear acceleration of its centre of mass. Compare with a hollow sphere of the same mass and radius.
Solution — Step by Step
Three forces act: gravity down, normal perpendicular to incline, and friction up the incline. Friction is what enables rolling without slipping.
Taking down the incline as positive:
Friction is the only force producing torque about the centre. With and :
For rolling without slipping, , so . Then .
Substituting into the force equation:
For a solid sphere, , so :
For a hollow sphere, :
vs . The solid sphere accelerates faster because more of its mass is closer to the axis, so less rotational kinetic energy is “wasted” — more energy goes into translation.
Final answer: , .
Why This Works
The factor tells us how “rotation-heavy” a body is. A point particle with would accelerate at — the full slide value. Any rolling body is slower because some of the gravitational energy converts to rotational KE.
Memorise these factors: ring , disc , solid sphere , hollow sphere . They appear in every rolling question in JEE.
Alternative Method
Energy conservation gives the same result for speed after rolling distance . Using and , we get , which combined with recovers our answer.
Common Mistake
Many students forget that friction does no work during pure rolling. The point of contact is instantaneously at rest, so the displacement of the friction force is zero. Energy is conserved entirely between gravitational PE, translational KE and rotational KE — friction is just the constraint that links them.