Three indentical uniform blocks of mass m each and length L are placed on a smooth fixed horizontal surface as shown. There is friction between A and B (Friction cofficient `mu`) while there is no friction between A and C. At the instant shown, that is at t=0, the block A has horizontal velocity of magnitude u towards right, whereas speed of B and C is zero. At the instant block A has covered a distance L relative to block B velcity of all blocks are same.

The magnitude of total work done by friction on system of three blocks is

Options

A. `-1/3 m u^(2)`
B. `-1/4 m u^(2)`
C. `-2/3 m u^(2)`
D. `1/2 m u^(2)`

Explanation

From conservation of momentum
`3mv="mu"` or `v=(u)/(3)`
Net workdone by friction `=(1)/(2)3m((u)/(3))^(2)-(1)/(2)"mu"^(2)=-(1)/(3)"mu"^(2)`
net work done by fricktion `=underset(r)overset(0)intmu(xlambdag)(-dx)=-mulambdag(L^(2))/(2)`
Also magnitude of net work done by friction `=mulambdag(L^(2))/(2)=mumg(L)/(2)`
`therefore(1)/(3)"mu"^(2)=mumg(L)/(2)` or `mu=(2)/(3)(u^(2))/(gL)`
`3mg="mu"` or `v=(u)/(3)`