Motion Of Centre Of Mass: Comprehensive NEET Physics Notes

1. Motion of Centre of Mass

1.1 Definition and Equation for Centre of Mass

The centre of mass of a system is the point where the entire mass of the system is considered concentrated for the purpose of analyzing translational motion. This concept simplifies the study of extended systems by focusing on a single point that represents the system’s mass distribution.

For a two-particle system, the position of the centre of mass is given by:

$X_{cm}=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}$

For a system of multiple particles, the centre of mass is expressed as:

$\mathbf{R}cm=\frac{\sum {i=1}^n{m}_i\mathbf{r}i}{\sum {i=1}^n{m}_i}$

Here:

• ${\mathbf{R}}_{cm}$ is the position vector of the centre of mass.
• $m_i$ is the mass of the i-th particle.
• ${\mathbf{r}}_i$ is the position vector of the i-th particle.

The centre of mass provides an effective way to study both translational and rotational motions in complex systems.

Did You Know?
The centre of mass of the Moon-Earth system lies within the Earth, which is why the Earth "wobbles" as the Moon orbits around it.

1.2 Motion of the Centre of Mass

Once we know the centre of mass, its motion follows Newton's second law as if all the external forces act on a single particle located at the centre of mass. The motion of the centre of mass is determined only by external forces, as internal forces cancel each other out due to Newton's third law.

The equation of motion for the centre of mass is:

$M\mathbf{A}cm=\sum \mathbf{F}ext$

Where:

• $M$ is the total mass of the system.
• ${\mathbf{A}}_{cm}$ is the acceleration of the centre of mass.
• $\sum {\mathbf{F}}_{ext}$ represents the sum of all external forces acting on the system.

Thus, the centre of mass behaves like a particle under the influence of the system's external forces.

Real-life Application:
In a car crash, passengers move toward the centre of mass of the car upon impact. This helps engineers design safer vehicles by placing safety features around the predicted centre of mass.

1.3 Practical Example: Centre of Mass of a Projectile

Consider a projectile that explodes mid-flight into two fragments. The internal forces causing the explosion have no effect on the centre of mass because internal forces cancel out. The centre of mass continues to follow the same parabolic trajectory that the projectile would have followed if it had not exploded. This illustrates that external forces (such as gravity) determine the centre of mass motion, not internal interactions like explosions.

Common Misconception:

Many students think that internal forces within a system can change the centre of mass's trajectory. However, only external forces can affect the motion of the centre of mass.

1.4 Example Problem: Centre of Mass of a Two-Particle System

Let's calculate the centre of mass for two masses of 3 kg and 5 kg, positioned at 2 m and 4 m along the x-axis, respectively:

$X_{cm}=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}=\frac{3\times 2+5\times 4}{3+5}=\frac{6+20}{8}=3.25m$

Thus, the centre of mass is located at 3.25 meters along the x-axis.

Quick Recap:

• The centre of mass simplifies the analysis of systems by allowing us to treat a system of particles as a single point.
• The motion of the centre of mass is determined by external forces only.
• The position of the centre of mass for a system of particles is a mass-weighted average of the particles’ positions.
• Internal forces do not affect the motion of the centre of mass.

Practice Questions

1. Two masses of 4 kg and 6 kg are placed at coordinates (3, 0) m and (0, 6) m. Find the coordinates of the centre of mass.
2. A system consists of three masses: 2 kg at (1, 0) m, 3 kg at (3, 2) m, and 5 kg at (5, 4) m. Determine the position of the centre of mass.
3. A uniform rod of length 8 m is placed along the x-axis with one end at the origin. Find the centre of mass of the rod.
4. Explain how internal forces affect or do not affect the motion of the centre of mass.
5. A projectile breaks into two fragments mid-air. Discuss the motion of the centre of mass after the explosion.

NEET Exam Strategy:

• Understand how the centre of mass simplifies the analysis of complex systems.
• Pay attention to collision and explosion problems where internal forces act within a system, but the centre of mass still moves as per external forces.
• Practice solving questions involving different mass distributions and calculating the centre of mass to reinforce the concept.

Glossary:

• Centre of Mass (CM): The point representing the average position of the mass distribution of a system.
• Internal Forces: Forces that act between particles within a system and cancel each other out when calculating the centre of mass's motion.
• External Forces: Forces exerted on a system from the outside, which determine the movement of the system's centre of mass.

Supplementary Features:

• Mnemonic:

  "Mass Concentrates at Centre" – to remember that the centre of mass behaves as if all mass is concentrated at a single point.

• Quick Reference Guide:
• • Position of CM for two masses: $X_{cm}=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}$
  • Position of CM for multiple particles: $\mathbf{R}cm=\frac{\sum {i=1}^n{m}_i\mathbf{r}i}{\sum {i=1}^n{m}_i}$