Question
A solid uniform disc of mass and radius rolls without slipping down a smooth incline of angle . Find the linear acceleration of its centre of mass and the friction force acting on it. Take .
Solution — Step by Step
For rolling without slipping, both translation and rotation matter. Along the incline (taking down as positive):
About the centre, friction provides torque:
where for a solid disc.
No slipping means , so . Substitute:
Substitute back into the translational equation:
Plugging in: .
Final: and up the incline.
Why This Works
Rolling friction here is static, not kinetic — the contact point momentarily has zero velocity. That is why we cannot use blindly. Instead, friction adjusts itself so the no-slip condition is satisfied.
The factor in the acceleration is specific to a solid disc. For a ring, you would get , and for a solid sphere, . Toppers memorise these three.
Alternative Method
Energy conservation gives speed but not acceleration directly. Using the instantaneous-axis-through-contact-point trick:
Gravity provides the torque, friction does not (it acts at the axis). Solve to get the same .
Common Mistake
Students assume friction must point down the incline because the disc is moving downhill. Actually, friction here points up the incline — it is what gives the disc the angular acceleration needed to keep rolling without slipping.