Motion of a Rigid Body in Physics

Spin is ordinary motion seen from a different seat. Replace distance with angle, speed with angular speed, force with torque and mass with moment of inertia, and every linear equation you already know reappears in rotational form. The one genuinely new idea is that moment of inertia is not a property of the material alone but of where the mass sits relative to the chosen axis, which is why a metre rule swung from its end is three times harder to start turning than the same rule twirled about its centre. The blocks run through the rotational twins of the kinematics, then the torque and inertia relations, then the energy that spin stores, and finish with rolling, where translation and rotation are locked together.

Average angular speed

Given

(rad)
(s)

Instantaneous angular speed

Given

Average angular acceleration

Given

(rad/s)
(s)

Instantaneous angular acceleration

Given

Angular speed after a time

Given

(rad/s)
(rad/s²)
(s)

Angular displacement after a time

Given

(rad/s)
(rad/s²)
(s)

Angular speed from the angular displacement, without the time

Given

(rad/s)
(rad/s²)
(rad)

Arc length from the angle turned

Given

(m)
(rad)

Tangential speed from the angular speed

Given

(m)
(rad/s)

Tangential acceleration from the angular acceleration

Given

(m)
(rad/s²)

Radial acceleration from the angular speed

Given

(m)
(rad/s)

Torque of a force about an axis

Given

(m)
(N)
(rad)

Torque from the components of the force and the position

Given

Net torque and the angular acceleration

Given

(kg m²)
(rad/s²)

Moment of inertia of a system of particles

Given

(kg)
(m)

Moment of inertia of a continuous body

Given

Parallel axis theorem

Given

(kg m²)
(kg)
(m)

Moment of inertia of a solid cylinder or disc about its axis

Given

(kg)
(m)

Moment of inertia of a thin hoop or ring about its axis

Given

(kg)
(m)

Moment of inertia of a solid sphere about a diameter

Given

(kg)
(m)

Moment of inertia of a thin spherical shell about a diameter

Given

(kg)
(m)

Moment of inertia of a uniform rod about one end

Given

(kg)
(m)

Rotational kinetic energy of a rigid body

Given

(kg m²)
(rad/s)

Rotational kinetic energy written in terms of angular momentum

Given

(kg m²/s)
(kg m²)

Moment of inertia from the rotational kinetic energy

Given

(J)
(rad/s)

Total kinetic energy of a body that both translates and rotates

Given

(kg)
(m/s)
(kg m²)
(rad/s)

Work done by a constant torque

Given

(N m)
(rad)

Power delivered by a torque

Given

(N m)
(rad/s)

Work done by the net torque and the change in rotational kinetic energy

Given

(kg m²)
(rad/s)

Condition for rolling without slipping

Given

(m)
(rad/s)

Acceleration condition for rolling without slipping

Given

(m)
(rad/s²)

Distance travelled by a rolling body for a given angle

Given

(m)
(rad)

Speed at the bottom of a slope for a rolling body

Given

(m)
(kg m²)
(kg)
(m)

Constant used

Acceleration down a slope for a rolling body

Given

(rad)
(kg m²)
(kg)
(m)

Constant used