JEE PYQ: Units & Measurements - Question ID 44648ec4a4b8 (JEE Main 2021)

In experiments involving indirect measurements, the fractional error (also called relative error) in a computed quantity is determined by the errors in the directly measured quantities. For a function of multiple variables, the fractional error propagates according to the rules of error propagation in multiplication and division.

Given the formula for Young’s modulus: $Y = \frac{MgL^3}{4b d^3 \delta}$ where:

• $M$ = mass (kg)
• $g$ = acceleration due to gravity (taken as exact, so no error)
• $L$ = length of the bar (m)
• $b$ = breadth of the bar (m)
• $d$ = thickness of the bar (m)
• $\delta$ = depression (m)

Since $g$ is assumed to have no error, the fractional error in $Y$ is given by: $\frac{\Delta Y}{Y} = \frac{\Delta M}{M} + \frac{3\Delta L}{L} + \frac{\Delta b}{b} + \frac{3\Delta d}{d} + \frac{\Delta \delta}{\delta}$ This formula arises because:

• Errors in multiplication/division add in quadrature for random errors, but for maximum error (worst-case scenario), we add absolute fractional errors.
• Exponents (like $L^3$ and $d^3$) multiply the fractional error by the exponent (due to logarithmic differentiation).

Step 1: Convert all observed values to SI units and identify least counts (absolute errors).

Step 2: Apply the error propagation formula.

The fractional error in $Y$ is: $\frac{\Delta Y}{Y} = \frac{\Delta M}{M} + 3 \cdot \frac{\Delta L}{L} + \frac{\Delta b}{b} + 3 \cdot \frac{\Delta d}{d} + \frac{\Delta \delta}{\delta}$ Substitute the fractional errors from the table: $\frac{\Delta Y}{Y} = 0.0005 + 3(0.001) + 0.0025 + 3(0.0025) + 0.002$

Step 3: Compute the total fractional error.

Break it down:

• $3 \cdot \frac{\Delta L}{L} = 3 \times 0.001 = 0.003$
• $3 \cdot \frac{\Delta d}{d} = 3 \times 0.0025 = 0.0075$

Step 4: Match with the given options.

The computed fractional error is $0.0155$, which corresponds to Option B.

1. Unit Conversion Errors: Students often forget to convert all measurements to SI units (e.g., cm to m, mm to m). Always ensure consistency in units before computing errors.
2. Ignoring Exponents in Error Propagation: A common mistake is to overlook the effect of exponents (e.g., $L^3$ contributes $3 \times \frac{\Delta L}{L}$, not just $\frac{\Delta L}{L}$).
3. Least Count vs. Absolute Error: The least count is the smallest measurable division, which directly gives the absolute error ($\Delta x$). Do not confuse it with fractional error.
4. Assuming $g$ has Error: The question states $g$ is taken without error. Ignoring this leads to incorrect calculations.
5. Rounding Errors: Intermediate rounding (e.g., $0.0025$ to $0.003$) can lead to inaccuracies. Retain full precision until the final step.

Exam Tip: For error propagation in products/quotients, always use: $\frac{\Delta Y}{Y} = \sum \left| n_i \cdot \frac{\Delta x_i}{x_i} \right|$ where $n_i$ is the exponent of $x_i$ in the formula. This ensures you account for all contributions correctly.