Step 1: Understanding the Concept:
Mass of a body is defined as the product of its density and its volume.
In experimental physics, measurements are never perfectly exact and are accompanied by errors.
When two physical quantities are multiplied, the relative errors of the individual quantities are added to find the relative error of the result.
The absolute error can then be derived from the relative error and the nominal value of the product.
Step 2: Key Formula or Approach:
Formula: \(M = \rho \times V\).
Relative error rule for products: \(\frac{\Delta M}{M} = \frac{\Delta \rho}{\rho} + \frac{\Delta V}{V}\).
Absolute error: \(\Delta M = M \times \left( \frac{\Delta \rho}{\rho} + \frac{\Delta V}{V} \right)\).
Step 3: Detailed Explanation:
Given values:
Density, \(\rho = 20 \text{ g/cm}^3\) with absolute error \(\Delta \rho = 4 \text{ g/cm}^3\).
Volume, \(V = 10 \text{ cm}^3\) with absolute error \(\Delta V = 1 \text{ cm}^3\).
First, calculate the measured (nominal) value of the mass:
\[ M = \rho \times V = 20 \text{ g/cm}^3 \times 10 \text{ cm}^3 = 200 \text{ g} \]
Next, calculate the relative errors for density and volume:
Relative error in density = \(\frac{\Delta \rho}{\rho} = \frac{4}{20} = 0.2\).
Relative error in volume = \(\frac{\Delta V}{V} = \frac{1}{10} = 0.1\).
Now, calculate the total relative error in mass:
\[ \frac{\Delta M}{M} = 0.2 + 0.1 = 0.3 \]
Finally, determine the absolute error in mass, \(\Delta M\):
\[ \Delta M = M \times 0.3 = 200 \text{ g} \times 0.3 = 60 \text{ g} \]
The absolute error in mass is 60 gm, which corresponds to option (D).
Step 4: Final Answer:
The calculated mass is 200 g, and based on the addition of relative errors (20% and 10% respectively), the total error is 30% of the mass, which is 60 g.