Physics Study Guide/Momentum

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.

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