We can also define centre of mass in terms of vector. let rᵢ be the position vector of the ith particle and R be the position vector of the centre of mass, then
![Relation Between Position Vectors Of Particles and Centre Of Mass](https://mrmanojpandey.com/wp-content/uploads/2023/01/Centre-of-mass-1.jpg)
Centre Of Mass Of Rigid Continuous Bodies
For a real body which is a continuous distribution of matter, point masses are then differential mass elements dm and centre of mass is defined as
![Centre Of Mass Of Rigid Continuous Bodies](https://mrmanojpandey.com/wp-content/uploads/2023/01/Centre-of-mass-1-1.jpg)
If origin is the co-ordinate of centre of mass, then
∫x dm = ∫y dm =∫z dm = 0
Centre Of Mass Of A Uniform Thin Road
We consider a uniform thin rod AB of mass M and length L. The rod is held along x-axis with its end A at the origin whose co-ordinate is (0,0). Let the element is placed side by side of length dx at distance x from the origin, then
![Centre Of Mass Of A Uniform Thin Road](https://mrmanojpandey.com/wp-content/uploads/2023/01/Centre-of-mass-2.jpg)
![Centre Of Mass Of A Uniform Thin Road](https://mrmanojpandey.com/wp-content/uploads/2023/01/Centre-of-mass-3.jpg)
![Centre Of Mass Of A Uniform Thin Road](https://mrmanojpandey.com/wp-content/uploads/2023/01/Centre-of-mass-4.jpg)
Thus, it means that the centre of mass of thin rod AB lies at point (L/2,0) i.e Centre between its end A and B.
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