6.4. LINEAR MOMENTUM OF A SYSTEM OF PARTICLES

Let us recall that the linear momentum of a particle is defined as

$m = p v \ (6.12)$

Linear momentum (p) is defined as the product of mass (m) and velocity (v) of an object. Essentially, it quantifies how much motion the object has. The more massive an object is or the faster it moves, the greater its linear momentum.

Think of a train and a small car. The train, with its large mass and speed, has a much larger momentum compared to the car, even if the car is moving fast. This helps to explain why a train is harder to stop than a car – it has more momentum.

Let us also recall that Newton’s second law written in symbolic form for a single particle is

dt = pF \ (6.13)

Newton's second law states that the force (F) acting on an object is equal to the rate of change of its momentum (p). This means that if you apply a force to an object, it will change its momentum over time. The greater the force you apply, the greater the change in momentum will be.

Imagine pushing a soccer ball. If you give it a light tap (small force), it moves slowly. But if you kick it hard (large force), it speeds off rapidly. The change in momentum corresponds directly to the force applied.

Let us consider a system of n particles with masses m1, m2,...mn respectively and velocities v1, v2,...vn respectively. The linear momentum of the first particle is m1v1, of the second particle is m2v2 and so on.

For the system of n particles, the linear momentum of the system is defined to be the vector sum of all individual particles of the system,

$P = p1 + p2 + ... + pn \ (6.14)$

The total linear momentum (P) of a system of particles is the sum of the individual linear momenta of all the particles within that system. Each particle contributes to the overall momentum based on its mass and velocity. This summation accounts for the direction of the velocities as well.

Consider a basketball team where each player contributes to the team's overall performance. If each player is trying to score points (momentum) at varying speeds, the total effectiveness of the team is the sum of each player's efforts (linear momentum).

Comparing this with Eq. (6.8),

$M = P V \ (6.15)$

Thus, the total momentum of a system of particles is equal to the product of the total mass of the system and the velocity of its centre of mass.

This equation states that the total momentum of a system can also be described as the mass of the system multiplied by the velocity of its centre of mass (V). The centre of mass behaves as if all the mass of the system were concentrated at that point when considering the momentum.

Think of a group of children pushing a sled. When they all push together in unison, the sled moves as if they all were a single entity at a single point – the centre of mass – which represents the sled's movement in a straightforward manner.

Thus, when the total external force acting on a system of particles is zero, the total linear momentum of the system is constant.

This is the law of conservation of the total linear momentum of a system of particles.

The principle of conservation of momentum states that in the absence of external forces, the total momentum of a system remains unchanged. It highlights the idea that momentum lost by some particles in a system must be gained by others when no outside force acts.

Consider a tug-of-war between two teams. If neither team receives assistance from outside, their total strength (momentum) remains the same, regardless of how hard they pull. It can fluctuate between the teams, but without external interference, it won't change overall.

The vector Eq. (6.18a) is equivalent to three scalar equations,

$P_x = c_1, P_y = c_2, P_z = c_3 \ (6.18 b)$

This concept states that the total linear momentum can be broken down into three components: along the x, y, and z axes. Each component can be treated separately to analyze the total momentum in three-dimensional space, which is a standard approach when working with vectors.

Imagine throwing a ball. Its motion can be analyzed in three plains: horizontally, vertically, and depthwise into the room. Each of these directions (x, y, z) contributes to the ball's overall path (momentum).

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