JEE PYQ: Units & Measurements - Question ID 612892c8e868 (JEE Main 2022)

Core Formula & Concept:

The density $\rho$ of a cylindrical wire is given by the ratio of its mass $m$ to its volume $V$. For a cylinder of radius $r$ and length $l$, the volume is: $V = \pi r^2 l$ Thus, the density formula becomes: $\rho = \frac{m}{\pi r^2 l}$

When dealing with errors in measurements, the relative error (or percentage error) in a derived quantity like density is determined using the rules of error propagation. If a quantity $Q$ depends on measured variables $x, y, z$ as: $Q = k x^a y^b z^c$ where $k$ is a constant, then the maximum relative error in $Q$ is: $\frac{\Delta Q}{Q} = |a| \frac{\Delta x}{x} + |b| \frac{\Delta y}{y} + |c| \frac{\Delta z}{z}$ This formula assumes the worst-case scenario where all errors add up constructively.

Step-by-Step Derivation:

Step 1: Express density in terms of measured quantities
Given: $\rho = \frac{m}{\pi r^2 l}$ Here, $\pi$ is a constant and does not contribute to error. The variables with errors are $m, r, l$.

Step 2: Compute relative errors in each measured quantity
We are given: - Mass: $m = 0.6 \pm 0.006$ g → $\frac{\Delta m}{m} = \frac{0.006}{0.6} = 0.01$ or $1\%$ - Radius: $r = 0.5 \pm 0.005$ mm → $\frac{\Delta r}{r} = \frac{0.005}{0.5} = 0.01$ or $1\%$ - Length: $l = 4 \pm 0.04$ cm → $\frac{\Delta l}{l} = \frac{0.04}{4} = 0.01$ or $1\%$

Step 3: Apply error propagation formula
Rewrite $\rho$ in the form suitable for error propagation: $\rho = \frac{m}{\pi r^2 l} = \frac{1}{\pi} \cdot m^1 \cdot r^{-2} \cdot l^{-1}$ Using the error propagation rule: $\frac{\Delta \rho}{\rho} = |1| \frac{\Delta m}{m} + |-2| \frac{\Delta r}{r} + |-1| \frac{\Delta l}{l} = \frac{\Delta m}{m} + 2 \frac{\Delta r}{r} + \frac{\Delta l}{l}$

Step 4: Substitute the relative errors
$\frac{\Delta \rho}{\rho} = 0.01 + 2(0.01) + 0.01 = 0.01 + 0.02 + 0.01 = 0.04$ Convert to percentage: $\frac{\Delta \rho}{\rho} \times 100\% = 4\%$

Step 5: Conclusion
The maximum percentage error in the measurement of density is $4\%$, which corresponds to option A.

Common Traps & Exam Tip:

Trap 1: Ignoring the exponent in error propagation. Many students forget that the error in $r^2$ is $2 \frac{\Delta r}{r}$, not just $\frac{\Delta r}{r}$. This leads to underestimating the error in density.

Trap 2: Miscounting the number of variables. Some students treat $\pi$ as a variable with error, but it is a constant and does not contribute to error.

Trap 3: Using absolute errors instead of relative errors. Always convert absolute errors to relative errors before applying the error propagation formula.

Exam Tip: When in doubt, write the formula in the form $Q = k x^a y^b z^c$ and apply the error propagation rule systematically. This avoids confusion and ensures accuracy.