Physics · JEE Main · JEE Advanced · NEET

Rotational Motion: concepts, formulas & solved PYQ-style examples

Rotational motion is the chapter most students find hardest in mechanics, and it is also one of the most tested. Most papers include a question on rolling, moment of inertia or angular momentum conservation.

The good news: nearly every question reduces to three tools — τ = Iα about the right axis, conservation of angular momentum when external torque is zero, and energy conservation for rolling. Pick the right tool first and the algebra is short.

Also for AP Physics 1Also for AP Physics C: MechanicsAlso for A-level Physics
Scan a Rotational Motion doubtJump to solved examples

Key concepts

Moment of inertia

Standard results for ring, disc, rod, sphere, plus the parallel- and perpendicular-axis theorems.

Torque and τ = Iα

Choose the axis carefully — about the contact point or the centre of mass for rolling bodies.

Angular momentum

Conserved when net external torque about the axis is zero: a person on a turntable, a lump sticking to a disc.

Rolling without slipping

v = ωR at the contact point; kinetic energy splits into translation and rotation.

Rolling on an incline

Acceleration g sinθ / (1 + k²/R²) — the body with the smaller k²/R² reaches the bottom first.

Formula sheet

Torque

τ=Iα=rFsin⁡θ\tau = I\alpha = rF\sin\theta

Angular momentum

L=Iω,τext=0⇒I1ω1=I2ω2L = I\omega,\quad \tau_{ext}=0 \Rightarrow I_1\omega_1 = I_2\omega_2

Parallel axis theorem

I=Icm+Md2I = I_{cm} + Md^2

Rolling kinetic energy

K=12Mv2(1+k2R2)K = \tfrac{1}{2}Mv^2\left(1+\tfrac{k^2}{R^2}\right)

Speed at bottom of incline

v=2gh1+k2/R2v = \sqrt{\dfrac{2gh}{1+k^2/R^2}}

Common k²/R²

ring 1, disc 12, solid sphere 25, hollow sphere 23\text{ring }1,\ \text{disc }\tfrac12,\ \text{solid sphere }\tfrac25,\ \text{hollow sphere }\tfrac23

What PYQs keep asking

Patterns that repeat across previous-year JEE and NEET papers for this chapter.

  • Comparing which body (ring, disc, sphere) reaches the bottom of an incline first or with more speed.
  • Angular momentum conservation when mass is added to or moved on a rotating body.
  • Moment of inertia of composite bodies using the parallel- and perpendicular-axis theorems.
  • Rolling with friction: finding the friction force and checking whether it acts up or down the incline.

Solved examples

Example 1

A solid sphere rolls without slipping down an incline from rest, dropping a height of 1.4 m. Find its speed at the bottom (g = 9.8 m/s²).

  1. 1For a solid sphere, k2/R2=25k^2/R^2 = \tfrac{2}{5}.
  2. 2Energy conservation: Mgh=12Mv2(1+25)=710Mv2Mgh = \tfrac12 Mv^2\left(1+\tfrac25\right) = \tfrac{7}{10}Mv^2.
  3. 3So v=10gh7=10×9.8×1.47=19.6v = \sqrt{\tfrac{10gh}{7}} = \sqrt{\tfrac{10 \times 9.8 \times 1.4}{7}} = \sqrt{19.6}.
Answer: v≈4.43v \approx 4.43 m/s
Watch a whiteboard video solution of this example →

Example 2

A uniform disc (mass 4 kg, radius 0.5 m) spins freely at 12 rad/s about its axis. A 1 kg lump of clay is dropped onto it and sticks at the rim. Find the new angular speed.

  1. 1Initial moment of inertia: I1=12MR2=12(4)(0.5)2=0.5I_1 = \tfrac12 MR^2 = \tfrac12(4)(0.5)^2 = 0.5 kg m².
  2. 2After the clay sticks: I2=0.5+mR2=0.5+1(0.5)2=0.75I_2 = 0.5 + mR^2 = 0.5 + 1(0.5)^2 = 0.75 kg m².
  3. 3No external torque about the axis, so I1ω1=I2ω2⇒ω2=0.5×120.75I_1\omega_1 = I_2\omega_2 \Rightarrow \omega_2 = \tfrac{0.5 \times 12}{0.75}.
Answer: ω2=8\omega_2 = 8 rad/s
Watch a whiteboard video solution of this example →

Common traps

  • Using v = ωR when the body is slipping — it holds only for rolling without slipping.
  • Forgetting the rotational part of kinetic energy, which gives v = √(2gh) for every body.
  • Taking moment of inertia about the centre when the question’s axis passes through the edge or end.

Practise with

SUVAT / Kinematics CalculatorFree Body Diagram MakerKinematics & Projectile MotionElectrostaticsCurrent ElectricityRotational dynamics and rolling simulation280 Rotational Motion practice questions130 Center of Mass, Momentum & Collision practice questionsVideo solutions

FAQ

Which body reaches the bottom of an incline first when rolling?

The one with the smallest k²/R²: solid sphere (2/5), then disc (1/2), then hollow sphere (2/3), then ring (1). Mass and radius do not matter.

When is angular momentum conserved?

When the net external torque about the chosen axis is zero — internal forces such as clay sticking or a person walking on a turntable do not change it.

Is rotational motion important for NEET?

Yes. NEET usually tests moment of inertia, torque and rolling with direct formula-based questions, while JEE Advanced adds multi-concept problems.

Stuck on a Rotational Motion question?

Snap your PYQ or homework question and get a whiteboard video that solves it step by step — your first video needs no login.

Scan a doubtView plans