angular and rotation and orbit

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.

Circular Orbit Kinetic Energy

In a circular orbit of a satellite around a planet, kinetic energy is constant (same speed).

Circular Orbit Gravitational Potential Energy

In a circular orbit, gravitational potential energy is constant (same orbital radius).

Circular Orbit Angular Momentum

In a circular orbit, angular momentum mr is constant (no external torques).

Circular Orbit Total Mechanical Energy

In a circular orbit, total mechanical energy is constant (no external work, and no internal energy).

Elliptical Orbit Kinetic Energy

In an elliptical orbit of a satellite around a planet, kinetic energy is NOT constant (speed changes).

Elliptical Orbit Gravitational Potential Energy

In an elliptical orbit, gravitational potential energy is NOT constant (orbital radius changes).

Elliptical Orbit Angular Momentum

In an elliptical orbit, angular momentum mor is constant (no external torques).

Elliptical Orbit Total Mechanical Energy

In an elliptical orbit, total mechanical energy is constant (no external work, and no internal energy).

Escape Velocity

Escape velocity is the minimum speed necessary for an object on the surface of a planet to reach a position far away from the planet.

Potential Energy Near Surface

Near the surface of a planet, potential energy is mgh, where h= 0 at the lowest point of the motion.

Potential Energy Away from Surface

Away from the surface, potential energy is PE = -GmM/d, which has a negative value except when far from the planet.