Centre of Mass:

The centre of mass of an object is the point at which the entire mass of the object is considered to be concentrated. It is the average position of all the particles that make up the object. The concept of centre of mass is used to simplify the analysis of the motion of a system of particles.

Definition and Calculation:

The centre of mass can be calculated by considering the individual masses of the particles and their respective positions. Mathematically, the centre of mass is given by the formula:

x_cm = (m1x1 + m2x2 + m3x3 + ... + mnxn) / (m1 + m2 + m3 + ... + mn)
y_cm = (m1y1 + m2y2 + m3y3 + ... + mny+n) / (m1 + m2 + m3 + ... + mn)
z_cm = (m1z1 + m2z2 + m3z3 + ... + mnzn) / (m1 + m2 + m3 + ... + mn)

where x_cm, y_cm, and z_cm are the coordinates of the centre of mass in the x, y, and z directions respectively. The mi's represent the masses of the particles, and xi, yi, and zi represent their respective positions.

Properties of Centre of Mass:

1. The centre of mass of a system of particles is not affected by external forces. It remains at the same position unless acted upon by an external force.

2. The centre of mass lies on the line of action of an external force. This means that if a force is applied to a system, the centre of mass will move in the direction of the force.

3. The centre of mass of a symmetrical object lies on its axis of symmetry. For example, in a uniform rod, the centre of mass lies at the midpoint.

Torque and Angular Momentum:

Torque is the rotational analogue of force. It is the product of force and the perpendicular distance from the axis of rotation. Mathematically, torque is given by:

τ = r × F

where τ is the torque, r is the position vector, and F is the force vector.

Angular momentum is the rotational analogue of linear momentum. It is the product of moment of inertia and angular velocity. Mathematically, angular momentum is given by:

L = Iω

where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

The centre of mass is used to calculate the torque and angular momentum of a system of particles. The total torque acting on a system about the centre of mass is equal to the rate of change of angular momentum of the system about the centre of mass.

Summary:

The centre of mass is the average position of all the particles that make up an object. It simplifies the analysis of the motion of a system of particles. The centre of mass is not affected by external forces and lies on the line of action of an external force. Torque and angular momentum are important concepts in rotational motion, and the centre of mass is used to calculate them in a system of particles.