Question

A cubical block f wood of specific gravity 0.5 and chunk of concrete of specific gravity 2.5 are fastened together in the ratio of mass of wood to the mass of concrete which makes the combination to float with entire volume of the combination submerged in water is:
1) 1:5
2) 1:3
3) 3:5
4) 2:5

Answer 1

Hint: Archimedes principle will be applied for this problem.
Also we have density is the ratio of mass upon volume and specific gravity is given as the ratio of density of the object to the density of water.
Using above relations we will solve the given problem.

Complete step by step solution:
Let’s define Archimedes principle first.
Archimedes Principal says that: when a body is wholly or partially immersed in a fluid, the buoyant force which acts on the body is equal to the weight of the fluid displaced by the object or body immersed.
Now comes to the calculation part:
Let volume of block be V₁ and volume of concrete be V₂ and mass of block as m_b and that of concrete be m_c .
According to Archimedes Principal :
Weight of the two masses = buoyancy force exerted by the liquid
$ \Rightarrow ({V_1} \times 0.5 \times g) + ({V_2} \times 2.5 \times g) = ({V_1} + {V_2}) \times 1 \times g$ (Weight is given as mass $ \times $ gravitational acceleration; specific gravity of water is 1)
$\Rightarrow {V_1} \times 0.5 + {V_2} \times 2.5 = {V_1} + {V_2} $
$ \Rightarrow {V_2} \times 1.5 = {V_1} \times .5 $
$ \Rightarrow \dfrac{{{V_1}}}{{{V_2}}} = \dfrac{3}{1} $
 (Ratio of volumes comes out to be 3/1)
Mass = specific gravity $ \times $ volume
$ \Rightarrow \dfrac{{{m_b}}}{{{m_c}}} = \dfrac{{\rho \times {V_1}}}{{\rho \times {V_2}}}$ ($\rho $ is specific gravity of wood and concrete respectively)

$ \Rightarrow \dfrac{{{m_b}}}{{{m_c}}} = \dfrac{{0.5 \times 3}}{{2.5 \times 1}}$(We have V₁ =3V₂)
$ \Rightarrow \dfrac{{{m_b}}}{{{m_c}}} = \dfrac{3}{5}$ (Ratio of mass of wood to mass of concrete)

Option 3 is correct.

Note:Specific gravity has no units because it is the ratio of two densities so the numerator and denominator units cancels each other and it is a scalar quantity. By scalar quantity we means it has only magnitude and no direction .Specific gravity can also be found out by calculating ratio of weights of the substance and water.