Masses m, ( 1 2 ) 1 2 m, ( 1 2 )^2 1 3 m ( 1 2 )^ N-1 1 N m are placed at x=1,2,3, N, respectively. If the total mass is M then the centre of mass of the system is

Step-by-step solution

According to given distribution of masses, centre of mass is given as aligned & X_ CM = m₁ x₁+m₂ x₂+m₃ x₃+ . .+m_n x_n M & m 1+ ( 1 2 ) m 2 2+ ( 1 2 )^2 m 3 3+ . . & = ( 1 2 )^ N-1 m N N+ M & = m+ ( 1 2 ) m+ ( 1 2 )^2 m+ + ( 1 2 )^ N-1 m + M & = m M [1+ 1 2 + ( 1 2 )^2+ .+ ( 1 2 )^ N-1 + . ] & = m M 1 1-1 / 2 & = m M 1 1 / 2 = 2 m M & aligned ( sum of infinite series in G. P, S = a 1-r ) = m M 1 1 / 2 = 2 m M Since, masses are distributed only along X -axis, hence Y_ CM =0 and Z_ CM =0 Position of centre of mass =