Centre of mass

Centre of mass

Introduction : We know that all the object can be considered to be made up of many small particles. When the body is at rest all the particle in that body is at rest and on the other hand when the body is in motion all the particle of the body is in motion. We have already studied that Newton's law is applicable for a point mass but in real sense the object in our surrounding is not a point mass. The concept of centre of mass help us to understand that in any system there exist a point which act as a point object for whole body and thus we are able to apply newton's law. For any rigid body the sum of moment of force of all the particle is equal to moment of force of a single point called as centre of mass. Let us study this topic in details.

Centre of mass : Centre of mass of any system is the point where all the mass of the system is concentrated. Let's understand this concept with the help of diagram given below.

From the above fig, Consider a system  made up of different particle having mass m_1,m₂,  m_n, at a distance of  r₁ , r₂,  r_n from Origin  ' O ' .Let M be the centre of mass of the system situated at a distance R from the point O.

The moment of  force for each particle is given by equation ,

m₁r₁ ; m₂r₂ ; m₃r₃ ; m_nr_n

The total moment of  force due to each particle is given by equation ,

m₁r₁  + m₂r₂ + m₃r₃  +m_nr_n------(1)

The moment of  force due to centre of mass is given by equation ,

MR ------(2)

From (1) and (2)

MR = m₁r₁  + m₂r₂ + m₃r₃  +m_nr_n

R = [(m₁r₁  + m₂r₂ + m₃r₃  +m_nr_{n) / M]---(A)}

_{Equation can be written as}

_{Equation A gives the location at which the centre of mass of a system is located.}

_{│<<<Torque│    │Velocity of CM>>>│}

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