Question

The mass of an object is measured as \( (28 \pm 0.01) \) g and its volume as \( (5 \pm 0.1) \) cm\(^3\). What is the percentage error in density?

• A. 1.20 %
• B. 0.35 %
• C. 2.04 %
• D. 0.71 %

Hint:

When calculating the percentage error in density, use the formula for error propagation for division, which adds the relative errors in mass and volume.

Answer 1

The Correct Option is C

Given:

• Mass of the object: \( m = 28 \pm 0.01 \, \text{g} \)
• Volume of the object: \( V = 5 \pm 0.1 \, \text{cm}^3 \)

Step 1: Formula for Density

The density \( \rho \) is given by the formula: \[ \rho = \frac{m}{V} \] where: - \( m \) is the mass, - \( V \) is the volume.

Step 2: Calculate the Relative Errors

The relative error in a calculated quantity is the sum of the relative errors in the measured quantities. So for density, the percentage error is the sum of the percentage errors in mass and volume. The relative error in mass is: \[ \frac{\Delta m}{m} = \frac{0.01}{28} = 0.000357 \] The relative error in volume is: \[ \frac{\Delta V}{V} = \frac{0.1}{5} = 0.02 \]

Step 3: Total Percentage Error in Density

The total percentage error in density is the sum of the percentage errors in mass and volume: \[ \text{Percentage Error in Density} = \left(\frac{\Delta m}{m} + \frac{\Delta V}{V}\right) \times 100 \] Substituting the values: \[ \text{Percentage Error in Density} = (0.000357 + 0.02) \times 100 = 2.0357 \% \]

Step 4: Final Answer

Rounding the result to two decimal places: \[ \boxed{2.04 \%} \]