Torque And Angular Momentum System Of Particles And Rotational Motion
Torque and Angular Momentum  System of Particles and Rotational Motion
Torque

Definition: Torque is the turning or twisting force that acts on an object around an axis or pivot.

Expression for Torque: $$ \tau= rF\sin\theta$$ Where:

$\tau$ is torque (measured in Newtonmeters, Nm)

r is the distance from the axis of rotation to the point where the force is applied (in meters, m)

F is the force applied (in Newtons, N)

$\theta$ is the angle between the force vector and the position vector from the axis of rotation (measured in radians)
Angular Momentum

Definition: Angular momentum is the measure of the rotational motion of an object and is defined as the product of the moment of inertia and the angular velocity.

Properties:

It is a vector quantity, both in direction and magnitude.

For a spinning object, the direction of angular momentum is given by the righthand rule.

The SI unit of angular momentum is kilogram meter squared per second (kg m^2/s)

Angular Momentum has both Orbital and Spin contribution

Relation between Torque and Angular Momentum: $$\overrightarrow{\tau}=\frac{d\overrightarrow{L}}{dt}$$ Where:

$\overrightarrow{\tau}$ is torque (measured in Newtonmeter rad)

$\overrightarrow{L}$ is angular momentum (measured in Kg. m2/s)

t is time (measured in seconds)
Conservation of Angular Momentum:

Statement: The total angular momentum of a closed system remains constant, regardless of the internal interactions within the system.

Conservation of Angular Momentum makes analysis and predictions easier. When external torque is absent:
$$I_i \omega_i =I_f \omega_f$$
 Applications:
 Ice skaters spinning faster when they pull their arms in.
 Conservation of angular momentum of planets orbiting the Sun.
 Actionreaction forces that occur during interactions.
Rotational Motion of a Rigid Body

Angular Velocity and Angular Acceleration

Angular Velocity($\omega$): The rate at which an object rotates. It is the change in angle per unit of time. $$\omega = \frac{d\theta}{dt}$$

Angular Acceleration($\alpha$): The rate at which angular velocity changes is known as angular acceleration. $$\alpha = \frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}$$

Equations of Rotational Motion: $$\omega_f=\omega_i+\alpha t$$ $$\theta_f=\theta_i+\omega_i t+\frac{1}{2} \alpha t^2$$

Kinetic Energy of Rotation: $$K=\frac{1}{2}I\omega^2$$

Moment of Inertia(I):

The measure of an object’s resistance to angular acceleration around a given axis.

Dependent upon
 Mass distribution
 Axis chosen

Parallel and Perpendicular Axes Theorem:

The moment of inertia about an axis perpendicular to a surface equals the sum of the moments of inertia of that surface about two mutually perpendicular axes in the plane of the surface, passing through the point at which the perpendicular axis intersects the plane.

The moment of inertia about an axis parallel to an axis through the center of mass is equal to the moment of inertia about the parallel axis plus the product of the total mass and the square of the distance between the axes.

Moment of Inertia for Some Shapes:
Shape  Formula for Moment of Inertia (I) 

Uniform solid sphere (radius R)  ( \frac{2}{5} MR^2) 
Thin, uniform hoop (mass M, radius R)  ( MR^2) 
Thin, uniform disk/cylinder (mass M, radius R)  ( \frac{1}{2} MR^2 ) 
Long uniform rod (mass M, length L) rotating about an axis perpendicular to the rod and through its center of mass  ( \frac{1}{12} ML^2) 
Long, uniform rod rotating about an axis perpendicular to the rod and passing through one end  ( \frac{1}{3} ML^2) 