Rotational Motion

Angular velocity is rate of change of angle. Angular acceleration is rate of change of angular velocity.

Rotational analogues of the three equations of motion. Valid for constant angular acceleration.

Linear velocity, tangential acceleration, and centripetal acceleration in terms of angular quantities.

Moment of inertia is the rotational analogue of mass. Measures resistance to angular acceleration.

MI about any axis equals MI about parallel axis through CM plus Md². d = distance between axes.

For a planar body: MI about axis perpendicular to plane equals sum of MIs about two perpendicular axes in the plane.

All mass at distance R from axis.

Uniform solid disc about its central axis.

Thin uniform rod about centre: ML²/12. About one end: ML²/3.

Uniform solid sphere about any diameter.

Thin spherical shell about any diameter.

Uniform solid cylinder about its symmetry axis. Same as disc.

Thin cylindrical shell about its symmetry axis. Same as ring.

Uniform rectangular plate of sides a and b about central perpendicular axis.

Turning effect = force × perpendicular distance of its line of action from the pivot.

Torque is the rotational analogue of force. It is the moment of force about an axis.

Net torque equals moment of inertia times angular acceleration. Rotational analogue of F = ma.

Angular momentum of a particle or rigid body. Rotational analogue of linear momentum.

Angular momentum is conserved when no net external torque acts on the system.

Net torque equals rate of change of angular momentum. Rotational analogue of F = dp/dt.

Kinetic energy of a rotating body. Rotational analogue of ½mv².

Sum of translational and rotational KE. k = radius of gyration.

For a body rolling without slipping, contact point has zero velocity relative to ground.

Acceleration of a body rolling without slipping down an incline of angle θ. k = radius of gyration.

Speed at the bottom of an incline of height h for a body rolling from rest.

Work done by a torque through angular displacement θ. Rotational analogue of W = Fd.

Power delivered by a torque at angular velocity ω. Rotational analogue of P = Fv.

Radial distance from the axis of rotation at which the entire mass can be concentrated without altering the moment of inertia.

Moment of inertia of a uniform circular disc of mass M and radius R about any diameter in its plane.