A block `A` os mass `m` slides on a smooth slider in the system as shown. A block `c` of mass hanging from a pulley pulls block `A`. When the block `A` was at position `B`, the spring was unstretched. Find the speed of the block `A` when `AB = OB = L`.
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Options

A. `[(gL)/(sqrt(2))-(KL^(2) sqrt(2))/(m)]^((1)/(2))`
B. `[gL -(KL^(2))/(2m) (sqrt(2) -1)^(2)]^((1)/(2))`
C. `[gL -(2KL^(2))/(m)(sqrt(2) -1)^(2)]^((1)/(2))`
D. `[(gL)/(2)-(KL^(2) sqrt(2))/(m)]^((1)/(2))`

Answer

B

Explanation

Decrease in `PE` = Increase in `KE` + Increase in elastic `PE`
`mgL=(1)/(2)xx(2m)v^(2)+(1)/(2)Kx^(2)=mv^(2)+(1)/(2)KL^(2)(sqrt(2)-1)^(2)`
`v=[gL-(KL^(2))/(2m)(sqrt(2)-)^(2)]^((1)/(2))`.