Question

A crown made of gold and copper weights $210g$ in air and $198g$ in water. The weight of gold in the crown is: (given: density of gold = $19.3gm/cm^3$ and density of copper = $8.5gm/cm^3$.)
A) 93g
B) 100g
C) 150g
D) 193g

Answer 1

Hint: Weight is defined as the force exerted on a body by gravity. Density is defined as a mass of a unit volume of a substance. Buoyancy is the force that causes the objects to float. It is caused due to the differences in pressure that are acting on the opposite side of the object which is immersed in a static fluid.

Complete step by step solution:
Given data:
Weight of crown in air = $210 g$
Weight of crown in water = $198 g$
The density of gold = $19.3 gm/cm^3$
The density of copper = $8.5 gm/cm^3$
Let the volume of gold be $V_g$ and the volume of copper be $V_c$
$ \Rightarrow $ $19.3 V_g $ + $8.5 V_c = 210$ -------- (1 )
The Buoyancy force in water results in a loss of weight of 12 gm
$ \Rightarrow $Buoyancy force, $B = 12 gm \times g$
$ \Rightarrow $Buoyancy force, B $ = $$\left( {{V_g} + {V_c}} \right)g = 12g$
$ \Rightarrow $$\left( {{V_g} + {V_c}} \right) = 12$-------- $($2 )
On solving the two equations, we get
$V_g = 10$ and $V_c = 2$
Thus the weight of gold $ = $10 $ \times $19.3 $ = $193 gm

Hence the correct option is D.

Note: 1) Buoyancy can also be defined as the upward force applied by the fluid on the object or the body when an object is kept in or submerged in the fluid.
2) Density is a derived quantity. It is defined as a measure of the amount of matter which is packed into a measured volume.
3) Weight is a force acting on the matter.