Rotational Motion formulas for JEE Main & Advanced

22 formula entries for Rotational Motion, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Angular Displacement, Velocity & Acceleration

ω = \frac{d θ}{d t}

α = \frac{d ω}{d t}

When to use. Describing rotational kinematics of a rigid body.

Trap. Forgetting direction: use right-hand rule for vector nature.

Rotational Kinematics Equations

ω = ω_{0} + α t

;

θ = ω_{0} t + \frac{1}{2} α t^{2}

;

ω^{2} = ω_{0}^{2} + 2 α θ

When to use. Problems with constant torque or angular acceleration.

Trap. Using these only when $α$ is constant; not for variable acceleration.

Moment of Inertia (I)

I = \sum m_{i} r_{i}^{2}

I = \int r^{2} d m

When to use. Calculating rotational kinetic energy, torque, angular momentum.

Trap. Assuming $I$ is same for all axes; it changes with axis.

Parallel Axis Theorem

I = I_{c m} + M d^{2}

When to use. Find $I$ about an axis offset from center of mass.

Trap. Using it for axes not parallel; only for parallel axes.

Perpendicular Axis Theorem

I_{z} = I_{x} + I_{y}

When to use. Find $I$ of a lamina about an axis perpendicular to its plane.

Trap. Only applicable for planar bodies (2D).

Standard Moments of Inertia

I_{ring} = M R^{2}

I_{disc} = \frac{1}{2} M R^{2}

I_{sphere} = \frac{2}{5} M R^{2}

I_{rod,center} = \frac{1}{12} M L^{2}

I_{rod,end} = \frac{1}{3} M L^{2}

When to use. Direct substitution in rotational dynamics problems.

Trap. Mixing up disc and ring formulas.

Torque ( $τ$ )

\vec{τ} = \vec{r} \times \vec{F}

∣ τ ∣ = r F \sin θ

When to use. Calculating rotational effect of a force.

Trap. Using $r$ as distance from axis, not perpendicular distance.

Newton's Second Law for Rotation

τ_{net} = I α

When to use. Problems with angular acceleration due to torques.

Trap. Forgetting to take net torque, not just one torque.

Angular Momentum ( $L$ )

\vec{L} = \vec{r} \times \vec{p}

(particle),

L = I ω

(rigid body)

When to use. Conservation of angular momentum in collisions or when torque is zero.

Trap. Using $L = I ω$ for particles; only for rigid bodies about axis.

Conservation of Angular Momentum

I_{1} ω_{1} = I_{2} ω_{2}

(if

τ_{ext} = 0

)

When to use. Problems where shape or axis changes without external torque.

Trap. Applying when external torque is present (e.g., friction).

Rotational Kinetic Energy

K_{rot} = \frac{1}{2} I ω^{2}

When to use. Energy conservation in rotational systems.

Trap. Forgetting to add translational KE for rolling objects.

Rolling Without Slipping

v_{cm} = R ω

K = \frac{1}{2} M v_{cm}^{2} + \frac{1}{2} I_{cm} ω^{2}

When to use. Problems with wheels, balls, cylinders rolling on surfaces.

Trap. Assuming friction is always present; pure rolling can occur without friction on horizontal surface.

Work-Energy Theorem for Rotation

W = \int τ d θ = \frac{1}{2} I ω_{f}^{2} - \frac{1}{2} I ω_{i}^{2}

When to use. Problems with variable torque or angular displacement.

Trap. Forgetting that torque must be about the axis of rotation.

Pulley Problems with Rotation

τ = (T_{1} - T_{2}) R = I α

a = R α

When to use. Problems with massive pulleys and hanging masses.

Trap. Assuming tensions on both sides are equal; they differ due to pulley's inertia.

Radius of Gyration ( $k$ )

k = \sqrt{\frac{I}{M}}

When to use. Comparing rotational inertia of different bodies.

Trap. Confusing $k$ with actual radius; it's a mathematical concept.

Equilibrium of Rigid Bodies

\sum \vec{F} = 0

\sum \vec{τ} = 0

When to use. Problems with ladders, beams, or objects in balance.

Trap. Using torque equations about a point without including all forces, or forgetting to also satisfy $\sum \vec{F} = 0$ .

Angular Impulse

\int τ d t = Δ L = I Δ ω

When to use. Problems with collisions or sudden torques.

Trap. Using this only when torque is constant; for variable torque, integrate.

Rotational Collisions

L_{initial} = L_{final}

When to use. Problems where a particle hits a rotating body or two rotating bodies collide.

Trap. Forgetting to include angular momentum of both bodies.

Torque as a Vector

\vec{τ} = \vec{r} \times \vec{F}

When to use. Calculating torque in 3D or when force is not perpendicular.

Trap. Order matters: $\vec{r} \times \vec{F} = - \vec{F} \times \vec{r}$ .

Relation Between Torque and Angular Momentum

\vec{τ} = \frac{d \vec{L}}{d t}

When to use. Deriving conservation or analyzing time-varying rotation.

Trap. Using this only for fixed axis; for variable axis, it's more complex.

Moment of Inertia of Composite Bodies

I_{total} = \sum I_{i}

When to use. Problems with multiple masses or shapes attached.

Trap. Forgetting to use same axis for all components.

Rolling on Inclined Plane

a = \frac{g \sin θ}{1 + \frac{I}{M R^{2}}}

When to use. Comparing rolling times or speeds of different objects.

Trap. Assuming all objects roll with same acceleration; shape matters.

Read the full chapter: Rotational Motion for JEE, with what JEE asks and the common traps.