A uniform cylinder of length L and mass M having cross - sectional area A is suspended , with its length vertical from a fixed point by a massless spring , such that it is half submerged in a liquid of density `sigma` at equilibrium position . The extension `x_(0)` of the spring when it is in equilibrium is .

Options

A. `(Mg)/k `
B. `(Mg)/k (1-(LAsigma)/M)`
C. `(Mg)/k (1-(LAsigma)/(2M))`
D. `(Mg)/k (1+(LAsigma)/M)`

Answer

C

Explanation

Mass of cross section A , `M = (AL)/2 ` and Length `L/2 `
` :. ` Buoyant force in liquid of sensity `sigma `
`F_(B) = igma Mg = sigma (Alg)/2 `
Suppose extension of spring is `x_(0)`
` :. ` Restoring force in upward direction in spring ` = kx_(0)`
When k = spring constant and force of weight on the cylinder in downward direction = Mg
` :. ` For equilibrium condition of cylinder ,
` :. mg = (sigmaALg)/2 + kx_(0)`
` :. kx_(o)=mg - (sigmaALg)/2 `
` :. x_(0) =(mg-(sigmaALg)/2)/k`
` :. x_(0) =(Mg)/k (1- (sigmaLA)/(2M))`