Question

A piece of iron of density \[7.8\,g/c{m^3}\] and volume \[100\,c{m^3}\] is fully immersed in water. Calculate
(a) Weight of iron piece in air
(b) Upthrust of water on iron piece
(C) Apparent weight of iron piece in water (\[g = 10\,m/{s^2}\])

Answer 1

Hint: To find the weight of the iron piece, first we need to find its mass by multiplying its density and volume, and then getting the weight by using the acceleration due to gravity. To find the buoyancy, we need to use the density of water and its volume, and use the product with the gravitational acceleration. The apparent weight is the subtraction of the two.

Complete step by step answer:
Let \[\rho \] be the density of the iron piece. Thus, the value of \[\rho \] is \[\rho = 7.8\,g/c{m^3}\] . Now, let $V$ be the volume of the iron piece. Thus, \[V = 100c{m^3}\]. We know the relationship between the mass of an object with the volume and density of the object as follows:
\[M = V\rho \]
Here, $M$ is the mass of an object. Thus, the mass of the iron piece would be:
\[
M=V\rho \\
\Rightarrow M=100\centerdot 7.8=780g \\
\Rightarrow M=0.78kg \\
\]
The weight of any object is given by the product of the mass of that object and the acceleration due to gravity. Thus weight is given as:
\[
W=Mg \\
\Rightarrow W=0.78\centerdot 10=7.8N \\
\]
Here, W is the weight and g is the acceleration due to gravity.
Upthrust of water is nothing but the buoyancy force acting on the iron piece because of the water displaced by the iron piece as it gets submerged. Thus, as we know the formula of buoyancy, the upthrust can be shown as:
\[{F_B} = {\rho _w}Vg\]
Here, \[{\rho _w}\]is the density of water and FB is the buoyancy force.
\[
{{F}_{B}}={{10}^{3}}\centerdot 100\times {{10}^{(-6)}}\centerdot 10 \\
\therefore {{F}_{B}}=1N
\]
The apparent weight of iron pieces in water can be obtained by subtracting buoyancy force from the true weight of iron pieces. Thus; apparent weight = $W-{F_B} = 6.8N$.

Note: The apparent weight is the weight which would be read on a spring balance or generally a weight balance and it will always be less than the true weight as there would be two forces acting on the iron piece, and both of them will be in the opposite direction. Hence apparent weight would always be less or equal to true weight.