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A spherical ball of 0.5 x 103 mm3 volume is subjected to a volumetric stress of 0.5 MPa. If bulk modulus of the bail material is 16 GPa, the change in volume of the ball would be approximately equal to
  • a)
    0.0156 mm3
  • b)
    1.56 x 10-4mm3
  • c)
    0.156 mm3
  • d)
    0.00156 mm3
Correct answer is option 'A'. Can you explain this answer?
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Bulk modulus
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Understanding the Problem
To find the change in volume of the spherical ball when subjected to a volumetric stress, we can utilize the concept of bulk modulus.
Given Data
- Volume of the ball (V) = 0.5 x 10^3 mm³
- Volumetric stress (σ) = 0.5 MPa = 0.5 x 10^6 Pa
- Bulk modulus (K) = 16 GPa = 16 x 10^9 Pa
Formula for Change in Volume
The change in volume (ΔV) can be expressed using the formula:
ΔV = - (σ / K) * V
Where:
- σ is the applied stress.
- K is the bulk modulus.
- V is the original volume.
Calculating the Change in Volume
1. Substituting Values:
- V = 0.5 x 10^3 mm³ = 0.5 x 10^-6 m³ (convert mm³ to m³ for consistency).
- σ = 0.5 x 10^6 Pa.
- K = 16 x 10^9 Pa.
2. Plugging into the Formula:
ΔV = - (0.5 x 10^6 / 16 x 10^9) * (0.5 x 10^-6)
3. Calculating:
- ΔV = - (0.03125) * (0.5 x 10^-6)
- ΔV = - 0.015625 x 10^-6 m³
- Converting back to mm³: ΔV ≈ -0.0156 mm³.
Conclusion
The negative sign indicates a decrease in volume, confirming that the ball's volume decreases under stress. Thus, the change in volume of the ball is approximately 0.0156 mm³, aligning with option 'A'.
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