JEE PYQ: Units & Measurements - Question ID 05f7c62d40f5 (JEE Main 2025)

Core Formula & Concept:

Density ($\rho$) of a cylindrical wire is defined as the mass per unit volume. For a wire of mass $m$, radius $r$, and length $L$, the volume $V$ is given by the formula for the volume of a cylinder:

$V = \pi r^2 L$ Therefore, the density is: $\rho = \frac{m}{V} = \frac{m}{\pi r^2 L}$ When dealing with errors in measurements, the relative error (or percentage error) in a derived quantity like density is found using the rules of error propagation. Specifically:

• For multiplication or division, the relative errors add up.
• For a quantity raised to a power (e.g., $r^2$), the relative error is multiplied by that power.

The formula for the maximum percentage error in density is: $\frac{\Delta \rho}{\rho} \times 100\% = \left( \frac{\Delta m}{m} + 2 \frac{\Delta r}{r} + \frac{\Delta L}{L} \right) \times 100\%$ Here, $\Delta m$, $\Delta r$, and $\Delta L$ are the absolute errors in mass, radius, and length, respectively.

Step-by-Step Derivation:

Given data:

• Mass, $m = 0.60 \pm 0.003 \text{ g}$
• Radius, $r = 0.50 \pm 0.01 \text{ cm}$
• Length, $L = 10.00 \pm 0.05 \text{ cm}$

Step 1: Compute the relative error in mass ($\frac{\Delta m}{m}$): $\frac{\Delta m}{m} = \frac{0.003}{0.60} = 0.005$ Step 2: Compute the relative error in radius ($\frac{\Delta r}{r}$): $\frac{\Delta r}{r} = \frac{0.01}{0.50} = 0.02$ Since radius is squared in the density formula, its relative error contribution is doubled: $2 \frac{\Delta r}{r} = 2 \times 0.02 = 0.04$ Step 3: Compute the relative error in length ($\frac{\Delta L}{L}$): $\frac{\Delta L}{L} = \frac{0.05}{10.00} = 0.005$ Step 4: Sum the relative errors to find the total relative error in density: $\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 2 \frac{\Delta r}{r} + \frac{\Delta L}{L}$ $\frac{\Delta \rho}{\rho} = 0.005 + 0.04 + 0.005 = 0.05$ Step 5: Convert the relative error to percentage error: $\text{Percentage error} = \frac{\Delta \rho}{\rho} \times 100\% = 0.05 \times 100\% = 5\%$ Thus, the maximum percentage error in the measurement of density is 5%.

Common Traps & Exam Tip:

Students often make the following mistakes in such questions:

1. Ignoring the power rule for errors: Many forget to multiply the relative error of the radius by 2 (since $r$ is squared in the volume formula). This leads to an underestimation of the total error.
2. Miscounting the number of terms: Some students mistakenly add the relative error of $\pi$ (which is zero, as $\pi$ is a constant with no error) or forget to include the error in length.
3. Incorrectly handling units: While the units of mass, radius, and length are consistent here (all in cm and g), students sometimes panic about unit conversion. In this problem, no conversion is needed as all units are compatible.
4. Rounding errors prematurely: It is crucial to keep intermediate calculations precise (e.g., $0.005$ instead of rounding to $0.01$) to avoid compounding errors.

Exam Tip: Always write down the formula for the derived quantity (here, density) and explicitly identify how each measured quantity contributes to the error. This systematic approach minimizes mistakes and ensures full credit in exams.