Question

A body of volume $100\;{\rm{c}}{{\rm{m}}^{\rm{3}}}$ weighs $5\;{\rm{kgf}}$ in air. It is completely immersed in a liquid of density $1.8 \times {10^3}\;{\rm{kg}}{{\rm{m}}^{{\rm{ - 3}}}}$. Find the weight of the body in liquid.
A. $4.82\;{\rm{kgf}}$
B. $9.64\;{\rm{kgf}}$
C. $2.41\;{\rm{kgf}}$
D. None of the above

Answer 1

The above problem can be resolved using the concepts and applications of the Buoyant force. Moreover, this problem is related to the Archimedes Principle, which says that a body immersed in the liquid will remove some quantity of liquid. That quantity will be equal to the weight of the body. Moreover, the mathematical relation for the weight of the body in the liquid is used, that is given by subtracting the real weight of the body and magnitude of upthrust acting on the body.

Complete step by step answer:
Given:
The weight of the body is, $W = 5\;{\rm{kgf}}$.
The density of the liquid is, $\rho = 1.8 \times {10^3}\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}}$.
The volume of body is,

V = 100\;{\rm{c}}{{\rm{m}}^{\rm{3}}}\\\ V = 100\;{\rm{c}}{{\rm{m}}^{\rm{3}}} \times \dfrac{{1\;{{\rm{m}}^{\rm{3}}}}}{{{{10}^9}\;{\rm{c}}{{\rm{m}}^{\rm{3}}}}}\\\ V = 100 \times {10^{ - 9}}\;{{\rm{m}}^{\rm{3}}} \end{array}$$ We know the expression for the weight of the body in liquid is, $${W_1} = W - U$$…...(1) Here, U is the upthrust acting on the body and it can be expressed as, $$U = V \times \rho \times g$$ here. g is the gravitational acceleration and its value is $$9.8\;{\rm{m/}}{{\rm{s}}^{\rm{2}}}$$. Solve by substituting the values as, $$\begin{array}{l} U = 100 \times {10^{ - 9}}\;{{\rm{m}}^{\rm{3}}} \times 1.8 \times {10^3}\;{\rm{kg/}}{{\rm{m}}^{\rm{3}}} \times 9.8\;{\rm{m/}}{{\rm{s}}^{\rm{2}}}\\\ U \approx 0.18\;{\rm{kgf}} \end{array}$$ On substituting the above result in equation 1, we get, $$\begin{array}{l} {W_1} = 5\;{\rm{kgf}} - 0.18\;{\rm{kgf}}\\\ {W_1} = 4.82\;{\rm{kgf}} \end{array}$$ Therefore, the weight of the body in the liquid is $$4.82\;{\rm{kgf}}$$ and option (A) is correct. **Note:** To resolve the given problem, one must understand the concept and applications of the Archimedes principle and also remember some mathematical relations, including the relation for the weight of the body in the liquid. The Archimedes principle has various applications, including the analysis of the hydraulic engines and turbomachinery.