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Question: A body weighs \[150\,g\] in air, \[120\,g\] in water and \[130\,g\] in liquid. The density of the li...

A body weighs 150 g150\,g in air, 120 g120\,g in water and 130 g130\,g in liquid. The density of the liquid in gcm−3gc{m^{ - 3}} is:
A. 23\dfrac{2}{3}
B. 45\dfrac{4}{5}
C. 1315\dfrac{{13}}{{15}}
D. 1513\dfrac{{15}}{{13}}

Explanation

Solution

Density of a substance is given by the ratio of the mass of the substance to the volume of the substance. Mathematically the density of an object is given by, d=mVd = \dfrac{m}{V} where, is the mass of the object and is the volume of the object. The weight of an object is the measure of force acting on it due to gravitational acceleration.Mathematically the weight of a body is given by, W=mg=dgVW = mg = dgV, where the weight of the object g is the gravitational acceleration and is the volume of the object. From Archimedes’s principle we know the apparent weight of the body in a liquid can be written as, W=mg−dgVW = mg - dgV. Where, dd is the density of the liquid in which the body is placed

Complete step by step answer:
We know that, from Archimedes’s principle the relative weight of the body in a liquid can be written as, W=mg−dgVW = mg - dgV. Where, dd is the density of the liquid in which the body is placed.Now, when the body is in water the weight of the body is given as, Wwater=120g{W_{water}} = 120g. The weight of the body in air means true weight is m=150 gm = 150\,g. Hence, we can write, 120g=150g−1gV120g = 150g - 1gV. Where, gg is the gravitational acceleration and density of water (d)=1 gcm−3\text{density of water}\,(d)= 1\,gc{m^{ - 3}}. Simplifying we get,
120=150−1V120 = 150 - 1V
⇒V=30\Rightarrow V = 30
So, the volume of the body is V=30gcm−3V = 30gc{m^{ - 3}}.
Now, when the body is inside the liquid of an unknown densitydd, we can write,
130g=150g−dgV130g = 150g - dgV
Up on simplifying we get,
130=150−30d130 = 150 - 30d
⇒d=2030\Rightarrow d = \dfrac{{20}}{{30}}
∴d=23 gcm−3\therefore d = \dfrac{2}{3}\,gc{m^{ - 3}}

Hence, the correct answer is option A.

Note: The apparent weight of a body cannot be greater than the weight of the body in air. Since, the weight always acts downwards for a body. If the apparent weight was greater than the weight of the body then the body would float until it reaches the surface of the liquid.