Centre of Mass of a Rigid Body
The centre of mass of a rigid body is a point whose position is fixed with respect to the body as a whole. The point may or may not lie in the body. The position of the centre of mass of a rigid body depends on.
- Shape of the body
- Distribution of mass in the body
If we consider the body with continuous distribution of matter, the summation in the formula of centre of mass should be replaced by integration. Therefore, we do not talk of the ith particle, rather, we talk of a small element of the body having a mass dm. If x, y, z are the co-ordinates of this small mass dm, we write the coordinates of this small mass dm. We write the co-ordinates of the centre of mass as
The integration is to be performed under proper limits so that, as the integration variable goes through the limits, the smaller elements cover the entire body.
For rigid bodies having regular geometrical shapes and uniform distribution of mass, the centre of mass is at their geometrical centres.| Object | Position of the centre of mass |
| Uniform hollow sphere | Centre of the sphere |
| Uniform solid sphere | Centre of the sphere |
| Uniform circular ring | Centre of the ring |
| Uniform circular disc | Centre of the disc |
| Uniform rod | Centre of the rod |
| A plane lamina (Rectangular or square) | Point of intersection of diagonals |
| Triangular plane lamina | Point of intersection of medians of the triangle |
| Rectanqular or cubical block | Point of intersection of diagonals |
- Consider a ceiling fan in your room. When it is on, each point of its body goes in a circle. If we locate the centres of the circles traced by different particles on the three blades of the fan and the body covering the motor, they all lie on a vertical line through the centre of the body. The fan rotates about this vertical line. This line is called the axis of rotation.
If we calculate the distance travelled by the different particles in a fan during the time-period of rotation of the fan, they are different. It increases as we move from the axis of rotation to the rim of the blades. But in a given time period, every particle of the body would have rotated through the same angle. This leads to the concept of angular velocity.
If the particles undergo a displacement (angular), during a small interval of time dt, then, the angular velocity 
