Question

In an experiment, a rectangular block with height h is allowed to float in two separate liquids. In the first liquid, which is water, it floats fully submerged. In the second liquid it floats with height h/7 above the liquid surface. What is the relative density (the density relative to that of water) of the second liquid?

Answer 1

Answer:

The relative density of the second liquid is 7.

Explanation:

From archimede's principle we know that the force that a liquid exerts on a object equals to the weight of the liquid that the object displaces.

Let us assume that the volume of the object is 'V'

Thus for the liquid in which the block is completely submerged

The buoyant force should be equal to weight of liquid

Mathematically

$F_{buoyant}=Weight\\\\\rho _{1}\times V\times g=m\times g\\\\\therefore \rho _{1}=\frac{m}{V}...............(i)$

Thus for the liquid in which the block is 1/7 submerged

The buoyant force should be equal to weight of liquid

Mathematically

$F'_{buoyant}=Weight\\\\\rho _{2}\times \frac{V}{7}\times g=m\times g\\\\\therefore \rho _{2}=\frac{7m}{V}.................(ii)$

Comparing equation 'i' and 'ii' we see that

$\rho_{2}=7\times \rho _{1}$

Since the first liquid is water thus $\rho _{1}=1gm/cm^3$

Thus the relative density of the second liquid is 7.

Answer 2

Answer:

7

Explanation:

Let the density of second liquid is d.

Density of water = 1 g/cm^3

In case of equilibrium, according to the principle of flotation, the weight of the body is balanced by the buoyant force acting on the body.

Let A be the area of cross section of block and D be the density of material of block and h be the height.

For first liquid (water):

Weight of block = m x g  = A x h x D x g .... (1)

Buoyant force in water = A x h x 1 x g  ..... (2)

Equating (1) and (2) we get

A x h x D x g = A x h x 1 x g

D = 1 g/cm^3

For second liquid:

Weight of block = m x g  = A x h x D x g .... (1)

Buoyant force in second liquid = A x h/7 x d x g  ..... (2)

Equating (1) and (2) we get

A x h x D x g = A x h/7 x d x g

D = d/7

d = 7 D = 7 x 1 = 7 g/cm^3

Thus, the relative density of second liquid is 7.