Centre of Mass Formula Sheet

Definition of Centre of Mass
The centre of mass of a system of particles is a point that represents the average position of all the particles in the system. It is a crucial concept in physics and helps in the analysis of various physical systems.

Centre of Mass Formula
The formula to calculate the centre of mass for a system of particles is given by:

x_cm = (m₁x₁ + m₂x₂ + m₃x₃ + ... + m_nx_n) / (m₁ + m₂ + m₃ + ... + m_n)

y_cm = (m₁y₁ + m₂y₂ + m₃y₃ + ... + m_ny_n) / (m₁ + m₂ + m₃ + ... + m_n)

Here, x_cm and y_cm represent the x-coordinate and y-coordinate of the centre of mass respectively. m₁, m₂, m₃, ..., m_n are the masses of the particles, and x₁, x₂, x₃, ..., x_n and y₁, y₂, y₃, ..., y_n are their corresponding x and y coordinates.

Centre of Mass of Regular Objects
For regular objects with uniform density, the centre of mass can be calculated using the following formulas:

- For a uniform rod of length L, the centre of mass is located at L/2 from one end.
- For a uniform disc, the centre of mass is at the geometric centre of the disc.
- For a uniform triangular plate, the centre of mass is at the intersection of the medians.
- For a uniform square or rectangular plate, the centre of mass is at the intersection of the diagonals.

Centre of Mass of Continuous Objects
For continuous objects, the centre of mass can be calculated using integration. The formula is given by:

x_cm = (1/M) ∫x dm

y_cm = (1/M) ∫y dm

Here, M represents the total mass of the object, and the integrals are taken over the entire object.

Conclusion
The centre of mass is an important concept in physics and helps in understanding the motion and stability of objects.