Question
A solid sphere of mass and radius rolls without slipping down an incline of angle . Find its acceleration along the incline. (JEE Main 2024 Shift 1 pattern.)
Solution — Step by Step
Along the incline, two forces act on the sphere: down the slope and friction up the slope (friction must point up because the sphere tends to slide down).
Friction provides the torque about the centre of mass. Normal force and gravity pass through the centre, so they contribute nothing.
For pure rolling, , so .
From the torque equation: .
Substituting into the translation equation:
Final answer:
Why This Works
The factor comes directly from the moment of inertia of a solid sphere (). For a hollow sphere (), the same algebra gives . For a ring or hollow cylinder, .
The pattern: . Memorise this — it cracks every “rolling down an incline” question instantly.
Alternative Method
Energy conservation gives velocity, not acceleration directly. But for time-of-descent questions, energy works faster:
So .
Common Mistake
Students forget to include rotational kinetic energy and write . This gives — the answer for a sliding (not rolling) block. JEE loves this trap because both answers look “clean”.
Rolling acceleration is always less than because part of the gravitational PE goes into spinning, not translating. If your answer comes out larger than , you have a sign error.