Question:
A metal cube has an edge length of 90 cm. If $2 \times 10^9$ N m$^{-2}$ of pressure (or stress) is required to reduce the edge length to 89.5 cm, then the bulk modulus of the metal is:
A metal cube has an edge length of 90 cm. If $2 \times 10^9$ N m$^{-2}$ of pressure (or stress) is required to reduce the edge length to 89.5 cm, then the bulk modulus of the metal is:
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For solids, volumetric strain is three times linear strain: $\Delta V/V = 3\Delta L/L$.
Updated On: Jul 18, 2026
- $1 \times 10^{11}$ N m$^{-2}$
- $2.5 \times 10^{10}$ N m$^{-2}$
- $9 \times 10^{11}$ N m$^{-2}$
- $1.2 \times 10^{11}$ N m$^{-2}$
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The Correct Option is D
Solution and Explanation
Step 1: Understanding bulk modulus definition.
Bulk modulus is defined as: \[ B = \frac{\text{stress}}{\text{volumetric strain}} \] For a cube, volumetric strain is related to linear strain by: \[ \frac{\Delta V}{V} = 3 \frac{\Delta L}{L} \]
Step 2: Calculating linear strain.
Initial length = 90 cm, final length = 89.5 cm: \[ \Delta L = -0.5 \, \text{cm} \] \[ \frac{\Delta L}{L} = \frac{-0.5}{90} = -\frac{1}{180} \]
Step 3: Calculating volumetric strain.
\[ \frac{\Delta V}{V} = 3 \times \left(-\frac{1}{180}\right) = -\frac{1}{60} \]
Step 4: Applying stress value.
Given stress: \[ P = 2 \times 10^9 \, \text{N m}^{-2} \]
Step 5: Calculating bulk modulus.
\[ B = \frac{2 \times 10^9}{1/60} = 2 \times 10^9 \times 60 = 1.2 \times 10^{11} \]
Step 6: Final conclusion.
\[ \boxed{1.2 \times 10^{11} \, \text{N m}^{-2}} \]
Bulk modulus is defined as: \[ B = \frac{\text{stress}}{\text{volumetric strain}} \] For a cube, volumetric strain is related to linear strain by: \[ \frac{\Delta V}{V} = 3 \frac{\Delta L}{L} \]
Step 2: Calculating linear strain.
Initial length = 90 cm, final length = 89.5 cm: \[ \Delta L = -0.5 \, \text{cm} \] \[ \frac{\Delta L}{L} = \frac{-0.5}{90} = -\frac{1}{180} \]
Step 3: Calculating volumetric strain.
\[ \frac{\Delta V}{V} = 3 \times \left(-\frac{1}{180}\right) = -\frac{1}{60} \]
Step 4: Applying stress value.
Given stress: \[ P = 2 \times 10^9 \, \text{N m}^{-2} \]
Step 5: Calculating bulk modulus.
\[ B = \frac{2 \times 10^9}{1/60} = 2 \times 10^9 \times 60 = 1.2 \times 10^{11} \]
Step 6: Final conclusion.
\[ \boxed{1.2 \times 10^{11} \, \text{N m}^{-2}} \]
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