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Obtain an expression for the velocity of centre of mass for n particles of system. |
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Answer» Solution :Consider a system of n particles. Let `vec(r_(1)),vec(r_(2)),vec(r_3),...vec(r_(n))` be the position vectors of the particles of MASSES `m_(1),m_(2),m_(3),m_(n)` respectively w.r.t. the origin of a co-ordinate system. If `vecR` is the position vector of centre of mass, then `vec(r)_(cm)or vecR=(m_(1)vec(r_(1))+m_(2)vec(r_(2))+...m_(n)vec(r_(n)))/(m_(1)+m_(2)+...m_(n))` `vec(R)=(m_(1)vec(r_(1))+m_(2)vec(r_(2))+...m_(n)vec(r_(n)))/(M)` `therefore MvecR=m_(1)vec(r_(1))+m_(2)vec(r_(2))+...m_(n)vec(r_(n))""...(1)` Assuming that the mass of the system does not change with time differentiating (1) w.r.t. time, `M(dvecR)/(DT)=m_(1)(dvec(r_(1)))/(dt)+m_(2)(dvec(r_(2)))/(dt)+...m_(n)(dvec(r_(n)))/(dt)` but `(dvecR)/(dt)=vecV` VELOCITY of centre of mass, `(dvec(r_(1)))/(dt),(dvec(r_(2)))/(dt),....,(dvec(r_(n)))/(dt)` respectively are the velocities `vec(v_(1)),vec(v_(2)),...vec(v_(n))` of n particle. `therefore MvecV=m_(1)vec(v_(1))+m_(2)vec(v_(2))+...m_(n)vec(v_(n))""...(2)` is the velocity of centre of mass for given system. This formula can be written as ALSO, `vecV=(m_(1)vec(v_(1))+m_(2)vec(v_(2))+...m_(n)vec(v_(n)))/(m_(1)+m_(2)+...m_(n))` |
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