Physics Study Guide/Momentum

=Momentum=

Linear momentum
Momentum is equal to mass times velocity.

Angular momentum
Angular momentum of an object revolving around an external axis $$O$$ is equal to the cross-product of the position vector with respect to $$O$$ and its linear momentum.

Angular momentum of a rotating object is equal to the moment of inertia times angular velocity.

Force and linear momentum, torque and angular momentum
Net force is equal to the change in linear momentum over the change in time.

Net torque is equal to the change in angular momentum over the change in time.

Conservation of momentum
Let us prove this law.

We'll take two particles $$a,b$$. Their momentums are $$\vec p_a,\vec p_b$$. They are moving opposite to each other along the $$x$$-axis and they collide. Now force is given by:
 * $$\vec F=\frac{d\vec p}{\mathrm{d}t}$$

According to Newton's third law, the forces on each particle are equal and opposite.So,
 * $$\frac{d\vec{p}_a}{dt}=-\frac{d\vec{p}_b}{dt}$$

Rearranging,
 * $$\frac{d\vec{p}}{dt}+\frac{d\vec{p}_b}{dt}=0$$

This means that the sum of the momentums does not change with time. Therefore, the law is proved.

Variables

Calculus-based Momentum
Force is equal to the derivative of linear momentum with respect to time.

Torque is equal to the derivative of angular momentum with respect to time.