Question

A cube of metal is subjected to a hydrostatic pressure of 4GPa. The percentage change in the length of the side of the cube is close to: (Given bulk modulus of metal,\[B = 8 \times {10^{10}}Pa\]
A. 0.6
B. 20
C. 1.67
D. 5

Answer 1

Hint:When the fluid is in an equilibrium position, due to force of gravity a pressure is exerted on the fluid, which is known as hydrostatic pressure.The formula for bulk modulus of elasticity will be applied here. This formula is used when applying pressure. There is some change in volume. It is denoted by symbol B and represented by the units Newton per meter square.

Formula Used:
Mathematically, the Bulk modulus formula is given by,
\[B = \dfrac{{\Delta P}}{{\dfrac{{\Delta V}}{V}}}\]……(i)
Where ‘B’ is the bulk modulus of elasticity, \[\Delta P\] is the change in pressure per unit area, \[\Delta V\] is the change in volume and V is the original or initial volume.

Complete step by step solution:
Given that the pressure is \[\Delta P = 4\,GPa = 4 \times {10^9}Pa\].
Let the length of the side of the cube is ‘a’.
The volume of cube is given by the formula= \[{a^3}\]
Change in volume is given by,
\[\dfrac{{\Delta V}}{V} = 3 \times \dfrac{{\Delta a}}{a}\]
\[\Rightarrow \dfrac{{4 \times {{10}^9}}}{{8 \times {{10}^{10}}}} = 3 \times \dfrac{{\Delta a}}{a}\]
\[\Rightarrow \dfrac{{\Delta a}}{a} = \dfrac{1}{{60}}\]
Calculating the percentage change, we get
\[\dfrac{{\Delta a}}{a} \times 100 = \dfrac{1}{{60}} \times 100\]
\[\therefore \dfrac{{\Delta a}}{a} = 1.67\]
Therefore, the percentage change in the length of the side of the cube is equal to 1.67.

Hence, option C is the correct answer.

Note: It is important to note that the hydrostatic pressure varies with the depth of the liquid. As the depth increases, the pressure also increases. This is because as the depth increases, the bottom layer of the fluid experiences pressures from the top layer, which increases the pressure of the overall container. Therefore, at the bottom of the container, the pressure is maximum and it increases as the depth increases. Also, another factor is that the liquid exerts pressure in all directions.