First slide

Fluid statics

Question

Two non-mixing liquids of densities $ρ$ and $n ρ (n > 1)$ 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 $p L$ $(𝑝 < 1)$ in the denser liquid. The density $d$ is equal to

Moderate

A

${2 + (n - 1) p} ρ$
B

${1 + (n - 1) p} ρ$
C

${1 + (n + 1) p} ρ$
D

${2 + (n + 1) p} ρ$

Solution

$d = d e n s i t y o f c y l i n d e r A = a r e a o f c r o s s - s e c t i o n o f c y l i n d e r u \sin g l a w o f f l o a t a t i o n,$
Using law of floatation,
Weight of cylinder = Upthrust by two liquids
$\begin{array}{l} L \times A \times d \times g = n ρ \times (p L \times A) g + ρ (L - p L) A g \\ d = n p ρ + ρ (1 - p) = (n p + 1 - p) ρ \\ d = {1 + (n - 1) p} ρ \end{array}$

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