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Question: The density of ice is \(917{\text{ kg/}}{{\text{m}}^3}\) . What will be the fraction of the volume o...

The density of ice is 917 kg/m3917{\text{ kg/}}{{\text{m}}^3} . What will be the fraction of the volume of a piece of ice above water when it is floating in freshwater?
A. 0.0830.083
B. 0.0420.042
C. 0.4120.412
D. 0.8130.813

Explanation

Solution

When a body is partially or completely immersed in a fluid, a force acts on it in the upward direction which is called force of buoyancy or buoyant force. It is given by FB=σ×V×g{F_B} = \sigma \times V \times g where σ\sigma is the density of the body, VV is the volume of the body in consideration and gg is the acceleration due to gravity.
Due to this buoyant force, the body loses some of its weight which is equal to the weight of the fluid displaced by the immersed part of the body.

Complete step by step answer
As given in the question, the ice is partially immersed in water.
We know that when a body is partially or completely immersed in a fluid, a force acts on it in the upward direction which is called force of buoyancy or buoyant force. It is given by FB=σ×V×g{F_B} = \sigma \times V \times g where σ\sigma is the density of the body, VV is the volume of the body in consideration and gg is the acceleration due to gravity.
And according to the Archimedes’ Principle, due to this buoyant force, the body loses some of its weight which is equal to the weight of the fluid displaced by the immersed part of the body.
Let ff be the fraction of volume of the ice that is inside the water. Then (1−f)\left( {1 - f} \right) will be the fraction of volume above water. We know that the density of water ρ=1000 kg/m3\rho = 1000{\text{ kg/}}{{\text{m}}^3} and the density of ice is given σ=917 kg/m3\sigma = 917{\text{ kg/}}{{\text{m}}^3} . Now, we apply Archimedes’ Principle to find the answer i.e.
Buoyant Force = Displaced weight of water{\text{Buoyant Force }} = {\text{ Displaced weight of water}}
σ×(1−f)×V×g=ρ×f×V×g\sigma \times \left( {1 - f} \right) \times V \times g = \rho \times f \times V \times g
On substituting the value and simplifying we have
917×(1−f)=1000×f917 \times \left( {1 - f} \right) = 1000 \times f
On further solving we have
f=0.917f = 0.917
Therefore, the fraction of volume of the ice above water is (1−f)=1−0.917=0.083\left( {1 - f} \right) = 1 - 0.917 = 0.083

Hence, option A is correct.

Note: The Archimedes’ Principle has numerous important applications in our life. It is used in designing the ships, boats and other water bodies. They are also used in producing Hydrometers which are used to measure density of different liquids.