Rotational Motion
Angular kinematics, moment of inertia, torque, angular momentum, rolling motion, and rotational energy.
Angular Displacement, Velocity and Acceleration
→ DerivationAngular velocity is rate of change of angle. Angular acceleration is rate of change of angular velocity.
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Equations of Rotational Motion
→ DerivationRotational analogues of the three equations of motion. Valid for constant angular acceleration.
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Relation Between Linear and Angular Quantities
→ DerivationLinear velocity, tangential acceleration, and centripetal acceleration in terms of angular quantities.
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Moment of Inertia — Definition
→ DerivationMoment of inertia is the rotational analogue of mass. Measures resistance to angular acceleration.
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Parallel Axis Theorem
→ DerivationMI about any axis equals MI about parallel axis through CM plus Md². d = distance between axes.
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Perpendicular Axis Theorem
→ DerivationFor a planar body: MI about axis perpendicular to plane equals sum of MIs about two perpendicular axes in the plane.
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Moment of Inertia of a Rod
→ DerivationThin uniform rod about centre: ML²/12. About one end: ML²/3.
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Moment of Inertia of a Solid Sphere
→ DerivationUniform solid sphere about any diameter.
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Moment of Inertia of a Hollow Sphere
→ DerivationThin spherical shell about any diameter.
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Moment of Inertia of a Solid Cylinder
→ DerivationUniform solid cylinder about its symmetry axis. Same as disc.
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Moment of Inertia of a Hollow Cylinder
→ DerivationThin cylindrical shell about its symmetry axis. Same as ring.
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Moment of Inertia of a Rectangular Plate
→ DerivationUniform rectangular plate of sides a and b about central perpendicular axis.
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Torque
→ DerivationTorque is the rotational analogue of force. It is the moment of force about an axis.
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Newton's Second Law for Rotation
→ DerivationNet torque equals moment of inertia times angular acceleration. Rotational analogue of F = ma.
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Angular Momentum
→ DerivationAngular momentum of a particle or rigid body. Rotational analogue of linear momentum.
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Conservation of Angular Momentum
→ DerivationAngular momentum is conserved when no net external torque acts on the system.
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Torque as Rate of Change of Angular Momentum
→ DerivationNet torque equals rate of change of angular momentum. Rotational analogue of F = dp/dt.
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Rotational Kinetic Energy
→ DerivationKinetic energy of a rotating body. Rotational analogue of ½mv².
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Total KE of a Rolling Body
→ DerivationSum of translational and rotational KE. k = radius of gyration.
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Condition for Pure Rolling
→ DerivationFor a body rolling without slipping, contact point has zero velocity relative to ground.
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Acceleration of Rolling Body on Incline
→ DerivationAcceleration of a body rolling without slipping down an incline of angle θ. k = radius of gyration.
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Velocity of Rolling Body at Bottom of Incline
→ DerivationSpeed at the bottom of an incline of height h for a body rolling from rest.
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Work Done by Torque
→ DerivationWork done by a torque through angular displacement θ. Rotational analogue of W = Fd.
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Power in Rotation
→ DerivationPower delivered by a torque at angular velocity ω. Rotational analogue of P = Fv.
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