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

Provided data:

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

Step 1: Density Formula

Density (\( \rho \)) is calculated as: \[ \rho = \frac{m}{V} \] where \( m \) represents mass and \( V \) represents volume.

Step 2: Calculate Relative Errors

The relative error in a computed value is the sum of the relative errors of the measured components. For density, the percentage error is the sum of the percentage errors in mass and volume. The relative error for mass is: \[ \frac{\Delta m}{m} = \frac{0.01}{28} = 0.000357 \] The relative error for 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 individual percentage errors: \[ \text{Percentage Error in Density} = \left(\frac{\Delta m}{m} + \frac{\Delta V}{V}\right) \times 100 \] Substituting the calculated relative errors: \[ \text{Percentage Error in Density} = (0.000357 + 0.02) \times 100 = 2.0357 \% \]

Step 4: Final Result

The density's percentage error, rounded to two decimal places, is: \[ \boxed{2.04 \%} \]