Question

Two non-mixing liquids of densities ρ and nρ(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 pL(p < 1) in the denser liquid. The density d is equal to :-
(1) {1 + (n + 1)p}ρ
(2) {2 + (n + 1)p}ρ
(3) {2 + (n – 1)p}ρ
(4) 1 + (n – 1)p}ρ
PLZ EXPLAIN WITH SOLUTION

Answer 1

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Answer 2

Answer:

The density d is equal to {1 + (n - 1)p}ρ.

Explanation:

According to the law of floatation, the cylinder will float in the liquid if the weight of the liquid displaced by it is equal to its weight.

Given:

• $\rho$ = density of first liquid.

• $n\rho$ = density of second liquid.

• L = length of the cylinder.

• d = density of the cylinder.

Let the area of cross-section of the cylinder be A, such that, the weight of the cylinder is given by

$\rm W_c = Mass \times g\\=density \times Volume\times g\\=d\times AL\times g.$

g is the acceleration due to gravity.

Given that the cylinder floats with its axis vertical and length pL(p < 1) in the denser liquid.

The volume of the denser liquid in which the cylinder is immersed is given by

$\rm V_1 = Cross-sectional\ area\ of\ the\ cylinder \times height\ of\ the\ liquid\\ V_1 = A\times pL.$

The volume of the less denser liquid in which the cylinder is immersed is given by

$\rm V_2 = Cross-sectional\ area\ of\ the\ cylinder \times height\ of\ the\ liquid\\ V_2 = A\times (L-pL) = AL(1-p).$

The weight of the liquids displaced is given by

$\rm W_l = (Mass\ of\ first\ liquid\times g)+(Mass\ of\ second\ liquid\times g)\\W_l=(density\ of\ first\ liquid\times volume\ of\ first\ liquid\times g)+(density\ of\ second\ liquid\times volume\ of\ second\ liquid\times g)\\W_l=n\rho\times V_1\times g+\rho\times V_2 \times g\\=n\rho\times ApL\times g+\rho \times AL(1-p)\times g\\=n\rho ApLg+\rho AL(1-p)g\\=\rho ALg(np+1-p)\\=\rho ALg(1+(n-1)p).$

According to the law of floatation,

$\rm W_c = W_l \\d\times AL\times g=\rho ALg(1+(n+1)p)\\\Rightarrow d = \rho (1+(n-1)p).$

Thus, option (4) is correct.