Two non-mixing liquids of densities `rho` and `(n gt1)` are put in a container. The height of each liquid is `h`. A solid cylinder of length `L` and density `d` is put in this container. The cylinder floats with its axis vertical and length `pL(p lt 1)` in the denser liquid. The density `d` is equal to :

Options

A. `[1+(n+1)p]rho`
B. `[2+(n+1)p]rho`
C. `[2+(n-1)p]rho`
D. `[1+(n-1)p]rho`

Answer

D

Explanation

Let while floating, the length of solid cylinder in lilquid of density `n rho` id p L. The length of cylinder in liquid of density `rho` is L - p L `=(1-p)L`. Fig. 7(CF). 43.

Weight solid cylinder = AL d g Total upthrust on cylinder due to two liquids
`=(p L) A n rho g+(1 - p)LA rho g`
As cylinder is floating, so
weight of cylinder = total upthrust on cylilnder
`AL d g=(p L)A n rho g +(1- p)AL rho g`
or `d=pn rho+ (1- p)rho`
`=pn rho + rho- p rho=rho+(n-1)p rho`
`=rho[1+(n-1)p]`