Systems of Particles and Rotational Motion: Comprehensive NEET Physics Notes

1. Centre of Mass

1.1 Centre of Mass of a Two-Particle System

Formula:

$X = \frac{m _{1} x _{1} + m _{2} x _{2}}{m _{1} + m _{2}}$

Explanation:

$m_{1}, m_{2}$ : Masses of the two particles.
$x_{1}, x_{2}$ : Positions of the two particles along a straight line.
$X$ : Position of the center of mass.

The center of mass is the weighted average position of all the particles in the system. For two particles, it lies closer to the heavier particle.

Derivation: Consider two particles of masses $m_{1}$ and $m_{2}$ at positions $x_{1}$ and $x_{2}$ along a straight line. The center of mass is defined as:

$X = \frac{m _{1} x _{1} + m _{2} x _{2}}{m _{1} + m _{2}}$

This is derived by considering the mass-weighted average of the positions of the two particles.

Example Application:

Given $m_{1} = 2 kg$ at $x_{1} = 1 m$ and $m_{2} = 3 kg$ at $x_{2} = 2 m$ , the position of the center of mass is:

$X = \frac{2 \times 1 + 3 \times 2}{2 + 3} = \frac{2 + 6}{5} = \frac{8}{5} = 1.6 m$

1.2 Centre of Mass of a System of Particles

Formula:

$X = \frac{\sum _{i = 1}^{n} m _{i} x _{i}}{\sum _{i = 1}^{n} m _{i}}$

Explanation:

$m_{i}$ : Mass of the $i$ -th particle.
$x_{i}$ : Position of the $i$ -th particle.
$X$ : Position of the center of mass.

This formula generalizes the concept of the center of mass to any number of particles, taking the mass-weighted average of their positions.

Derivation: For a system of $n$ particles, the center of mass is:

$X = \frac{m _{1} x _{1} + m _{2} x _{2} + \dots + m _{n} x _{n}}{m _{1} + m _{2} + \dots + m _{n}} = \frac{\sum _{i = 1}^{n} m _{i} x _{i}}{\sum _{i = 1}^{n} m _{i}}$

Example Application:

Consider three particles with masses 1 kg, 2 kg, and 3 kg at positions 0 m, 1 m, and 2 m, respectively. The position of the center of mass is:

$X = \frac{1 \times 0 + 2 \times 1 + 3 \times 2}{1 + 2 + 3} = \frac{0 + 2 + 6}{6} = \frac{8}{6} = 1.33 m$

2. Linear Momentum and Center of Mass Motion

2.1 Linear Momentum of a System of Particles

Formula:

$P = M V$

Explanation:

$P$ : Total linear momentum of the system.
$M$ : Total mass of the system.
$V$ : Velocity of the center of mass.

The total linear momentum of a system of particles is equal to the product of the total mass of the system and the velocity of its center of mass.

Derivation: For a system of $n$ particles, the total linear momentum is given by:

$P = \sum_{i = 1}^{n} p i = \sum i = 1^{n} m_{i} v_{i} = M V$

where $M$ is the total mass and $V$ is the velocity of the center of mass.

Example Application:

If a system consists of two particles of masses 2 kg and 3 kg moving with velocities 1 m/s and 2 m/s respectively, the total linear momentum of the system is:

$P = 2 \times$