Linear Momentum of System — Definition

Definition

Imagine you're trying to stop a rolling bowling ball versus a rolling tennis ball. Which one is harder to stop? The bowling ball, right? That's because it has more 'oomph' or 'inertia in motion.' This 'oomph' is what we call linear momentum.

For a single object, linear momentum ($vec{p}$) is simply calculated by multiplying its mass ($m$) by its velocity ($vec{v}$). So, $vec{p} = mvec{v}$. Since velocity is a vector (it has both magnitude and direction), linear momentum is also a vector, pointing in the same direction as the velocity. The unit for linear momentum is kilogram-meter per second (kg·m/s).

Now, what if we have a group of objects, like a system of particles? Think of a collection of billiard balls on a table. The total linear momentum of this entire system is just the vector sum of the individual linear momenta of each ball.

So, if ball 1 has momentum $vec{p}_1$, ball 2 has $vec{p}_2$, and so on, the total momentum of the system ($vec{P}_{sys}$) would be $vec{P}_{sys} = vec{p}_1 + vec{p}_2 + vec{p}_3 + dots$. This total momentum can also be expressed in terms of the total mass of the system ($M$) and the velocity of its center of mass ($vec{v}_{CM}$), i.

e., $vec{P}_{sys} = Mvec{v}_{CM}$.

The most powerful concept related to linear momentum is its conservation. The Law of Conservation of Linear Momentum states that if no net external force acts on a system of particles, then the total linear momentum of the system remains constant.

This means that even if the particles within the system interact with each other (e.g., collide, push apart), their individual momenta might change, but their vector sum (the total momentum of the system) will not change.

These internal forces, like the forces between colliding billiard balls, always occur in action-reaction pairs and thus cancel each other out when considering the system as a whole. This law is incredibly useful for analyzing situations like collisions, explosions, and rocket propulsion, where external forces are either absent or negligible compared to the large internal forces involved during the interaction.