Momentum

Linear momentum

Momentum is equal to mass times velocity.

{\vec {p}}=m{\vec {v}}

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.

{\vec {L}}={\vec {r}}\times {\vec {p}}={\vec {r}}\times m{\vec {v}}

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

{\vec {L}}=I{\vec {\omega }}

Force and linear momentum, torque and angular momentum

Net force is equal to the change in linear momentum over the change in time.

{\vec {F}}={\frac {\Delta {\vec {p}}}{\Delta t}}

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

{\vec {\tau }}={\frac {\Delta {\vec {L}}}{\Delta t}}

Conservation of momentum

{\begin{matrix}{\vec {p}}_{i}={\vec {p}}_{f}\\{\vec {L}}_{i}={\vec {L}}_{f}\end{matrix}}

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

p: momentum, (kg·m/s)
m: mass, (kg)
v: velocity (m/s)
L: angular momentum, (kg·m²/s)
I: moment of inertia, (kg·m²)
ω: angular velocity (rad/s)
α: angular acceleration (rad/s²)
F: force (N)
t: time (s)
r: position vector (m)

Bold denotes a vector quantity.
Italics denotes a scalar quantity.

Definition of terms

Momentum (p): Mass times velocity. (kg·m/s)

Mass (m) : A quantity that describes how much material exists, or how the material responds in a gravitational field. Mass is a measure of inertia. (kg)

Velocity (v): Displacement divided by time (m/s)

Angular momentum (L): A vector quantity that represents the tendency of an object in circular or rotational motion to remain in this motion. (kg·m²/s)

Moment of inertia (I): A scalar property of a rotating object. This quantity depends on the mass of the object and how it is distributed. The equation that defines this is different for differently shaped objects. (kg·m²)

Angular speed (ω): A scalar measure of the rotation of an object. Instantaneous velocity divided by radius of motion (rad/s)

Angular velocity (ω): A vector measure of the rotation of an object. Instantaneous velocity divided by radius of motion, in the direction of the axis of rotation. (rad/s)

Force (F): mass times acceleration, a vector. Units: newtons (N)

Time (t) : (s)

Isolated system: A system in which there are no external forces acting on the system.

Position vector (r): a vector from a specific origin with a magnitude of the distance from the origin to the position being measured in the direction of that position. (m)

Calculus-based Momentum

Force is equal to the derivative of linear momentum with respect to time.

{\vec {F}}={\frac {d{\vec {p}}}{dt}}

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

{\vec {\tau }}={\frac {d{\vec {L}}}{dt}}

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