Centre of Mass: Comprehensive NEET Physics Notes

1. Centre of Mass: Introduction

The centre of mass (CM) of a system is the point where the entire mass of the system can be considered concentrated for the purpose of analyzing its motion. It is particularly useful when studying the motion of rigid bodies or systems of particles, simplifying complex systems into more manageable forms.

Consider a system of particles with masses $m_{1}, m_{2}, m_{3}, \dots, m_{n}$ at positions $r_{1}, r_{2}, r_{3}, \dots, r_{n}$ . The position of the centre of mass $R$ is given by:

$R = \frac{\sum m _{i} r _{i}}{\sum m _{i}}$

Where:

$R$ is the position vector of the centre of mass,
$m_{i}$ is the mass of the ith particle,
$r_{i}$ is the position vector of the ith particle.

If all particles lie along a straight line (1D system), the position of the centre of mass is:

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

1.1 Two-Particle System

For a simple two-particle system with masses $m_{1}$ and $m_{2}$ located at positions $x_{1}$ and $x_{2}$ , the centre of mass is given by:

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

If the two masses are equal, the centre of mass lies exactly midway between them.

1.2 General Formula for n Particles

For a system of $n$ particles in three-dimensional space, the centre of mass position can be calculated using:

$X = \frac{\sum _{i = 1}^{n} m _{i} x _{i}}{\sum m _{i}}, Y = \frac{\sum _{i = 1}^{n} m _{i} y _{i}}{\sum m _{i}}, Z = \frac{\sum _{i = 1}^{n} m _{i} z _{i}}{\sum m _{i}}$

Where $x_{i}, y_{i}, z_{i}$ are the coordinates of the ith particle.

Did You Know?

In rocket science, calculating the centre of mass helps in determining how a spacecraft moves after fuel consumption or stage separation.

2. Calculation of Centre of Mass for Common Objects

2.1 Centre of Mass of a Uniform Rod

For a uniform rod of length $L$ , the centre of mass lies at the geometric centre of the rod, which is at a distance $\frac{L}{2}$ from either end.

2.2 Centre of Mass of a System of Particles

For systems consisting of multiple particles, the centre of mass can be calculated using the general formula:

$X = \frac{\sum m _{i} x _{i}}{\sum m _{i}}, Y = \frac{\sum m _{i} y _{i}}{\sum m _{i}}, Z = \frac{\sum m _{i} z _{i}}{\sum m _{i}}$

For example, in a system of three particles at the vertices of an equilateral triangle, the centre of mass lies at the centroid of the triangle.

Real-life Application:

Gymnasts use the concept of centre of mass to perform flips and spins by adjusting their body position, allowing them to control balance and movement mid-air.

3. Properties of Centre of Mass

Motion of Centre of Mass: The motion of the centre of mass is affected only by external forces. Internal forces between particles do not influence the motion of the centre of mass.
Centre of Mass and Rigid Bodies: In rigid bodies, the centre of mass moves as if all the mass were concentrated at this point, and all external forces act through it.
Symmetry and Centre of Mass: For symmetrical objects with uniform mass distribution (such as spheres, discs, cylinders), the centre of mass coincides with the geometric centre of the object.

NEET Tip:

For NEET problems involving systems of particles or continuous objects, use symmetry to simplify calculations and save time during the exam.

4. NEET Problem-Solving Strategy for Centre of Mass

Identify the System: Clearly define the masses and positions of particles or objects.
Use the Formula: Apply the centre of mass formula for discrete particle systems or exploit symmetry for continuous systems.
Break into Components: For systems in 2D or 3D, calculate the x, y, and z coordinates of the centre of mass separately.

Common Misconception:

Students often believe that the centre of mass must always lie within the physical boundaries of the object, which is not true. For instance, the centre of mass of a hollow ring lies at the geometric centre, where there is no mass.

5. Centre of Mass in Various Shapes

5.1 Spherical Objects

For solid spheres or hollow shells, the centre of mass is located at the geometric centre. This is true for both uniform and symmetrical mass distributions.

5.2 Triangular Lamina

The centre of mass of a triangular lamina is located at the centroid, which can be found at the intersection of the medians of the triangle.

Mnemonic:

"Symmetry Simplifies Solutions" – Remember this when dealing with shapes like spheres, rods, or discs. If the shape is symmetrical, the centre of mass is likely at the geometric centre.

Quick Recap

Centre of mass simplifies the analysis of a system by treating i