Question

If the errors in the measurement of the mass and side of a cubical block are 2% and 1% respectively, then the error in the determination of the density of the material of the block is

• A. 8%
• B. 6%
• C. 3%
• D. 5%

Hint:

When calculating the error in derived quantities, use the formula for relative error propagation. For density, the error in volume (cube) contributes three times the error in the side length.

Answer 1

The Correct Option is D

The density \( \rho \) of a cubical block is given by: \[ \rho = \frac{{mass}}{{volume}} = \frac{m}{a^3} \] where \( m \) is the mass and \( a \) is the side length of the cube. The relative error in density is given by: \[ \frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 3 \cdot \frac{\Delta a}{a} \] Given: 
Error in mass \( \frac{\Delta m}{m} = 2% \) 
Error in side length \( \frac{\Delta a}{a} = 1% \)
Substituting the values: \[ \frac{\Delta \rho}{\rho} = 2% + 3 \cdot 1% = 2% + 3% = 5% \] Thus, the error in the determination of the density is 5%.
Final Answer:  5%