1. What is Linear Momentum of a System?
Linear momentum describes how difficult it is to stop a moving object. For a single particle, momentum is given by \( \vec{p} = m \vec{v} \). But real objects are made of many particles. To understand the behaviour of an entire body or system, we consider the total linear momentum of all its particles.
This total momentum gives us a single quantity that represents the collective motion of the entire system, even if individual particles move differently.
1.1. Why Do We Need System Momentum?
A football, a car, or even a box contains many particles. Analysing each particle separately is impossible. Instead, we combine all their momenta to understand how the whole object moves. This idea forms the foundation for understanding collisions, mechanical systems, and centre-of-mass motion.
1.1.1. Examples from Everyday Motion
- A moving bus has huge total momentum because of its large mass, even at moderate speed.
- A group of skaters pushing each other still moves together with a predictable total momentum.
- A rocket ejecting gas particles changes its total momentum to gain forward motion.
2. How Total Momentum is Calculated
The total momentum of a system is the vector sum of all momenta of its individual particles. If the system has particles with masses \( m_1, m_2, m_3, \ldots \) and velocities \( \vec{v}_1, \vec{v}_2, \vec{v}_3, \ldots \), then:
\( \vec{P} = \sum_i m_i \vec{v}_i \)
This expression simply adds up the momentum of each particle to form the total system momentum.
2.1. Relation with Centre of Mass Velocity
An important result in physics is that the total momentum of a system is related to the velocity of its centre of mass. If \( M = \sum m_i \) is the total mass of the system, then:
\( \vec{P} = M \vec{v}_{cm} \)
This means the entire system behaves like a single particle located at the centre of mass, moving with velocity \( \vec{v}_{cm} \).
3. Role of Internal and External Forces
The total momentum of a system changes only when external forces act. Forces inside the system cannot change the total momentum because they cancel out in pairs. Understanding this distinction helps explain why some systems maintain momentum even when internal motion is happening.
3.1. Internal Forces
Internal forces are the forces particles within the system exert on each other. According to Newton’s Third Law, these forces occur in action–reaction pairs. Therefore, the momentum change due to internal forces is zero.
3.2. External Forces
External forces act from outside the system. These forces determine how the total momentum changes. If no external force acts, the total momentum remains constant—even if internal changes occur.
4. Examples to Understand System Momentum
Total momentum helps explain several real-life and physics problems. Here are some common examples that illustrate how system momentum works in practical situations.
4.1. Colliding Billiard Balls
When two balls collide, internal forces act between them, but no external force exists. Hence, the total momentum of the system remains conserved.
4.2. Explosion of a Firecracker
When a firecracker breaks into pieces, internal forces cause the explosion. But since external forces are negligible, the total momentum of the system (all pieces) remains the same as before the explosion.
4.3. Rocket Propulsion
In a rocket, gases are pushed out at high speed. The backward momentum of the gases and the forward momentum of the rocket together keep the total momentum conserved when external forces are absent.
5. Summary of the Concept
The linear momentum of a system provides a powerful and simple way to describe the collective motion of multiple particles. By summing the momentum of all particles, we obtain a single vector quantity that makes analysis easier. Internal forces cancel out, and only external forces influence the change in momentum.
This concept becomes the foundation for the next topics, where we study motion of the centre of mass and the principle of conservation of momentum.