Rotational Motion for JEE: Moment of Inertia, Torque, and Angular Momentum Guide

The parallel axis theorem states: I = I_cm + Md², where I is the moment of inertia about any axis parallel to one through the center of mass, M is mass, and d is the distance between the two axes. It is essential for MOI calculations.

The perpendicular axis theorem applies to planar bodies: I_z = I_x + I_y, where I_z is the MOI about an axis perpendicular to the plane, and I_x, I_y are MOI about two perpendicular axes in the plane. Valid only for laminar bodies.

Angular momentum is conserved when the net external torque on the system is zero. This occurs in: (1) spinning ice skater pulling arms in, (2) planetary motion, (3) collisions where no external torque acts.

Radius of gyration (k) is defined by I = Mk². It is the distance from the axis where all mass would be concentrated to give the same moment of inertia. For a ring: k = R. For a solid sphere: k = √(2/5)R.

Mass is a measure of resistance to translational acceleration (F = ma). Moment of inertia is a measure of resistance to angular acceleration (τ = Iα). MOI depends on mass distribution relative to the axis.

Angular momentum L = Iω for rotational motion. For a particle: L = mvr = r × p. Direction is perpendicular to the plane of rotation (right-hand rule). L is conserved when net external torque = 0.

Work done by torque = change in rotational kinetic energy: W = τθ = Δ(½Iω²). This parallels the translational work-energy theorem: W = Fd = Δ(½mv²).

For a body rolling down an incline without slipping: a = g sinθ/(1 + k²/R²). Acceleration: solid sphere (k²/R² = 2/5): a = (5/7)g sinθ. Solid cylinder (1/2): a = (2/3)g sinθ. Ring (1): a = (1/2)g sinθ.

Pure rotation: every point moves in a circle about the axis, all points have same angular velocity. Pure translation: every point moves with same linear velocity. Rolling combines both: v = ωR.

Angular momentum is conserved (L = Iω constant). Pulling arms in decreases MOI, so angular velocity increases. The skater spins faster. This is angular momentum conservation in action.

The center of mass is the point where the entire mass can be considered concentrated. It moves as if all external forces act at this point. For torque problems, external forces can be considered to act at the COM.

A rigid body is in equilibrium when: (1) Net force = 0 (translational equilibrium), (2) Net torque = 0 (rotational equilibrium). Both conditions must be satisfied simultaneously.

Precession is the slow rotation of a spinning body's axis due to an external torque. Example: a spinning top precesses due to gravity. Gyroscopes use this principle. The precession rate ω_p = τ/L.