Relative Percentage Error in Density of Solid Ball

To find the relative percentage error in the density of the solid ball, we begin by writing the expression for density in terms of mass and diameter.

The density $\rho$ of a sphere of mass $m$ and diameter $D$ is given by:
$\rho = \frac{m}{V} = \frac{m}{\frac{4}{3}\pi \left(\frac{D}{2}\right)^3} = \frac{6m}{\pi D^3}$

The maximum relative error in density $\rho$ is expressed as:
$\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 3\frac{\Delta D}{D}$

Multiplying by $100\%$ gives the percentage error in density:
$\left(\frac{\Delta \rho}{\rho}\right) \times 100\% = \left(\frac{\Delta m}{m}\right) \times 100\% + 3\left(\frac{\Delta D}{D}\right) \times 100\%$

Step 1: Calculate the Least Count ($\text{LC}$) of the Screw Gauge

The least count of the screw gauge is determined by:
$\text{LC} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}} = \frac{0.5\text{ mm}}{50} = 0.01\text{ mm}$

Step 2: Determine the Measured Diameter ($D$) and Absolute Error ($\Delta D$)

The measured value of the diameter is:
$D = \text{Main Scale Reading (MSR)} + (\text{Circular Scale Reading (CSR)} \times \text{LC})$
$D = 2.5\text{ mm} + (20 \times 0.01\text{ mm}) = 2.5\text{ mm} + 0.20\text{ mm} = 2.70\text{ mm}$

The uncertainty/absolute error in measuring the diameter is equal to the least count of the instrument:
$\Delta D = \text{LC} = 0.01\text{ mm}$

Step 3: Calculate the Percentage Error in Diameter

$\left(\frac{\Delta D}{D}\right) \times 100\% = \frac{0.01\text{ mm}}{2.70\text{ mm}} \times 100\% = \frac{1}{2.7}\% = \frac{10}{27}\%$

Step 4: Calculate the Total Relative Percentage Error in Density

Given that the relative error in mass is $\left(\frac{\Delta m}{m}\right) \times 100\% = 2\%$:

$\left(\frac{\Delta \rho}{\rho}\right) \times 100\% = 2\% + 3 \times \left(\frac{10}{27}\%\right)$
$\left(\frac{\Delta \rho}{\rho}\right) \times 100\% = 2\% + \frac{10}{9}\%$
$\left(\frac{\Delta \rho}{\rho}\right) \times 100\% = 2\% + 1.11\% = 3.11\% \approx 3.1\%$

Thus, the relative percentage error in the density is $3.1\%$.

Correct Option: C