What is the formula for the center of mass Class 11?

The formula for calculating the center of mass depends on whether you're dealing with a system of discrete particles or a continuous mass distribution. Here's a breakdown of both, relevant for Class 11 physics:

Center of Mass for a System of Discrete Particles

When dealing with a system of individual, separate particles, the center of mass (COM) is calculated using the following formula:

R_CM = (m₁r₁ + m₂r₂ + ... + m_nr_n) / (m₁ + m₂ + ... + m_n) = (∑ m_ir_i) / (∑ m_i) = (∑ m_ir_i) / M

Where:

• R_CM is the position vector of the center of mass. This tells you the location of the center of mass relative to your chosen origin.
• m_i is the mass of the ith particle.
• r_i is the position vector of the ith particle. This describes the particle's location in space relative to your chosen origin.
• n is the total number of particles in the system.
• ∑ represents the summation (adding up) of the terms.
• M is the total mass of the system (M = m₁ + m₂ + ... + m_n).

This formula can be broken down into its components in Cartesian coordinates (x, y, z):

• X_CM = (m₁x₁ + m₂x₂ + ... + m_nx_n) / M
• Y_CM = (m₁y₁ + m₂y₂ + ... + m_ny_n) / M
• Z_CM = (m₁z₁ + m₂z₂ + ... + m_nz_n) / M

Center of Mass for a Continuous Mass Distribution

When dealing with an object where the mass is continuously distributed (like a solid object), the center of mass is calculated using integration:

R_CM = (∫ r dm) / (∫ dm) = (∫ r dm) / M

Where:

• R_CM is the position vector of the center of mass.
• r is the position vector of a small mass element dm.
• dm is an infinitesimally small mass element.
• ∫ represents the integral (summation over continuous values).
• M is the total mass of the object (M = ∫ dm).

In Cartesian coordinates, this becomes:

• X_CM = (∫ x dm) / M
• Y_CM = (∫ y dm) / M
• Z_CM = (∫ z dm) / M

To use this formula, you need to express dm in terms of the coordinates (x, y, z) and the density of the object. For example, if the object has uniform density ρ, then dm = ρ dV, where dV is a small volume element.

In summary, the appropriate formula for the center of mass depends on whether you're working with discrete particles or a continuous mass distribution. For discrete particles, a summation is used, while for continuous distributions, an integral is used.