Rolling Down an InclineJEE Advanced

Concept

A body rolling without slipping shares its energy between sliding and spinning. The more its mass sits far from the axis (larger $I/mR^2$), the more energy goes into rotation — so it accelerates slower. A solid sphere always wins a race against a ring.

Key formula

Derivation

Newton's law along the incline ($mg\sin\theta - f = ma$) and torque about the centre ($fR = I\alpha$, with $a = R\alpha$) combine to give $a = g\sin\theta/(1+k)$.

The shape factor $k$ is $2/5$ (solid sphere), $1/2$ (disc), $2/3$ (hollow sphere) and $1$ (ring). Smaller $k$ ⇒ larger acceleration and a smaller share of energy locked in rotation.

Scenarios to explore

• Rolling Down an Incline — Why a sphere beats a ring — I/mR² and energy split.

Real-world applications

• Predicting which object rolls down fastest.
• Wheel, gear and flywheel design.
• Analysing yo-yos and rolling coins.

JEE exam tips

• Acceleration order: sphere > disc > hollow sphere > ring (independent of mass and radius).
• Rolling friction does no work — mechanical energy is conserved.
• Minimum $\mu = \tan\theta\cdot k/(1+k)$ to avoid slipping.

Common mistakes

• Using $a = g\sin\theta$ (that's frictionless sliding, not rolling).
• Forgetting rolling needs friction to provide the torque.
• Mixing up $I$ with $I/mR^2$ in the formula.

Exam traps to avoid

• The result is independent of mass and radius — only the shape factor $k$ matters.
• Beyond the minimum $\mu$, the body rolls; below it, it slips and the formula breaks.