# Centre of Mass

### Introduction to Centre of Mass

• The centre of mass of a system of particles is the point that behaves as if all the mass were concentrated there.
• The centre of mass accounts for the locations and masses of all particles in a system.
• The motion of the system can be described purely by the motion of the centre of mass in both translational and rotational motion.
• For a uniform body in a symmetric shape, the centre of mass is located at the geometric centre.

### Mathematical Calculation of Centre of Mass

• The centre of mass can be determined using the following equations:
• For a one-dimensional system, x (centre of mass) = Σmx / Σm.
• For a two-dimensional or three-dimensional system, x (centre of mass) = Σmx / Σm, y (centre of mass) = Σmy / Σm, and z (centre of mass) = Σmz / Σm.
• In the above equations, m represents the mass of each particle and x, y and z represent their respective coordinates.

### Concepts Involving Centre of Mass

• The law of conservation of momentum relates closely to the centre of mass; the total momentum of a system remains constant unless acted on by an external force.
• Understanding the centre of mass is crucial for understanding rotational dynamics. The angular momentum of a rotating body remains constant unless acted on by an external torque.

### Centre of Mass and Stability

• The location of the centre of mass affects the stability of an object. An object is stable if a line drawn vertically through the centre of gravity falls within its base.
• Lowering the centre of mass, widening the base, or doing both can increase the stability of an object.

### Applications of Centre of Mass

• Understanding the centre of mass is vital in fields like biomechanics, athletics, vehicle design, and civil engineering. It helps in designing stable structures, understanding body movements in sports, and improving vehicle safety.

### Tips for Understanding Centre of Mass

• Practice computation of the centre of mass for different systems using the formulas mentioned above.
• Develop a thorough understanding of the relation between the centre of mass and concepts like momentum and stability.
• Analyse practical situations and relate them back to the theoretical understanding of the centre of mass.

In sum, the centre of mass plays a key role in studying the mechanics of a system of particles. Gaining a deep understanding of this concept not only aids in solving problems but can also lay a strong foundation for further learning in physics.