Rod `AO_(3)` of length `L` can rotate abput `A`. Initially rod was at position `AO_(2)`, when spring `OB` of force constant `K`, attached to block `B` of mass `m` was at position `OA` with unstretched length `L`. The smooth block `B` can slide on rod when pulled by the block `D` of mass `m` through a massless spring and smooth pulley at `O_(1)`.
Find the work done by the frictional force (if slider is rough) at the instant when rod and the spring attached at block `B` make an angle of `30^@` with their respective initial positions.

Options

A. `(1)/(2)KL^(2)(2- sqrt(3))^(2) -mgL`
B. `KL^(2)(2- sqrt(3))^(2)-(mgL)/(4)`
C. `(1)/(8)KL^(2)(2- sqrt(3))^(2) -(5)/(4) mgL`
D. `(1)/(2)KL^(2)(sqrt(2)-1)^(2)`

Answer

C

Explanation

`W = Delta KE = 0`
`W_(f) = W_(N) + W_(S) + W_(g) = 0`
`W_(N) =` Work done by normal reaction `= 0` (Arts perpendicular to displacement)
`W_(s)` = Work done by spring force
=`0 -(1)/(2) K(L-(Lsqrt(3))/(2))^(2)`
`W_(g)` = Work done by force of gravity `= (5)/(4) mgL`
`W_(f) =-[0-(1)/(2)KL^(2)((2-sqrt(3))/(2))^(2) + (5)/(4) mgL]`
` =(1)/(8) KL^(2) (2- sqrt(3))^(2) - (5)/(4) mgL`.