angular and rotation

Angular displacement

The angle through which an object has rotated, measured in radians.

Average angular velocity

Angular displacement divided by the time interval, measured in rad/s.

Instantaneous angular velocity

The rate at which an object is rotating at a specific moment in time.

Angular acceleration

The rate of change of an object's angular speed, measured in rad/s².

Centripetal acceleration

The acceleration that changes an object's direction of motion, independent of angular acceleration.

Linear displacement in rotational motion

Given by the equation x = rθ, where r is the distance from the rotational axis.

Linear speed in rotational motion

Given by the equation v = rω, where r is the radius and ω is the angular velocity.

Linear acceleration in rotational motion

Given by the equation a = rα, where r is the radius and α is the angular acceleration.

Torque

Given by τ = Fd, where F is the force applied and d is the lever arm.

Rotational inertia

An object's resistance to angular acceleration.

Rotational inertia for a point particle

Given by I = MR², where M is mass and R is the distance from the axis of rotation.

Parallel Axis Theorem

If the rotational inertia about the center of mass is known, then the rotational inertia about a parallel axis is I' = Icm + Md².

Angular momentum for extended object

Given by L = Iω, where I is rotational inertia and ω is angular speed.

Angular momentum for a point object

Given by L = mvr, where v is velocity and r is the distance of closest approach.

Conservation of angular momentum

When no external torques act on a system, the angular momentum remains constant.

Angular impulse-momentum theorem

ΔL = τΔt, indicating that a change in angular momentum equals the net torque multiplied by the time applied.

Rotational kinetic energy

Given by K = ½Iω², where I is rotational inertia and ω is the angular speed.