Question

Statement (A): For an ideal liquid, the bulk modulus is infinite and the shear modulus is zero.
Statement (B): The volume contraction of a metal cube of bulk modulus \(140\,\text{GPa}\), \(10\,\text{cm}\) on an edge, when subjected to a hydraulic pressure of \(7\times 10^6\,\text{Pa}\), is \(-0.05\,\text{m}^3\).
Statement (C): A spiral spring is stretched by a weight attached to it. The strain is tensile.

• A. A, B and C are true
• B. A, B are true, C is false
• C. A, C are true, B is false
• D. B and C are true, A is false

Hint:

For bulk modulus questions, use \(\Delta V=-\frac{PV}{K}\). Also, ideal liquids are incompressible and cannot resist shear stress.

Answer 1

The Correct Option is C

Step 1: Evaluate Statement A about ideal liquids.
An ideal liquid is defined as incompressible and non-viscous. Since it is incompressible, its volume does not change regardless of how much pressure is applied. By definition, bulk modulus $K = -V(dP/dV)$. If $dV = 0$, then $K \to \infty$. Also, an ideal liquid has no rigidity and cannot resist shear deformation, so its shear modulus is zero. Statement A is TRUE.
Step 2: Set up the calculation for Statement B.
Statement B claims the volume contraction is $-0.05 \, \text{m}^3$ for a metal cube with bulk modulus $K = 140 \, \text{GPa} = 140 \times 10^9 \, \text{Pa}$, edge length 10 cm = 0.1 m, under pressure $P = 7 \times 10^6 \, \text{Pa}$. The formula for volume contraction is: \[ \Delta V = -\frac{PV}{K} \]
Step 3: Calculate the volume of the metal cube.
Edge = 0.1 m, so: \[ V = (0.1)^3 = 10^{-3} \, \text{m}^3 \]
Step 4: Compute Delta V and check Statement B.
\[ \Delta V = -\frac{(7 \times 10^6)(10^{-3})}{140 \times 10^9} = -\frac{7 \times 10^3}{140 \times 10^9} = -5 \times 10^{-8} \, \text{m}^3 \] This is $-5 \times 10^{-8} \, \text{m}^3$, which is far from $-0.05 \, \text{m}^3$. The claimed value is wrong by a factor of about $10^6$. Statement B is FALSE.
Step 5: Evaluate Statement C about a spiral spring.
When a weight is attached to the bottom of a spiral spring and the spring stretches, its length increases. Strain = (change in length) / (original length). Since length increases, the strain is positive, which is classified as tensile strain. Statement C is TRUE.
Step 6: Identify the correct combination.
From the analysis: A is TRUE, B is FALSE, C is TRUE. This matches the option: A and C are true, B is false. \[ \boxed{\text{A and C are true, B is false}} \]