JEE PYQ: Units & Measurements - Question ID 99a39f47ef60 (JEE Main 2024)

A simple pendulum’s time period $T$ is related to its length $L$ and the acceleration due to gravity $g$ by the formula $T = 2\pi \sqrt{\frac{L}{g}}.$ Rearranging for $g$ gives $g = \frac{4\pi^2 L}{T^2}.$ When quantities are measured with uncertainties, the relative (percentage) uncertainty in $g$ is found by combining the relative uncertainties in $L$ and $T$ using the rules of error propagation for multiplication/division: $\frac{\Delta g}{g} = \frac{\Delta L}{L} + 2\,\frac{\Delta T}{T}.$ The percentage accuracy in $g$ is then $100 \times \frac{\Delta g}{g}$.

1. Identify measured values and their uncertainties
Length $L = 20\ \text{cm} = 0.20\ \text{m}$, with accuracy $\Delta L = 2\ \text{mm} = 0.002\ \text{m}$.
Time for 50 oscillations $t_{50} = 40\ \text{s}$, with resolution $\Delta t_{50} = 1\ \text{s}$.
Therefore the period of one oscillation is $T = \frac{t_{50}}{50} = \frac{40}{50} = 0.80\ \text{s},$ and its uncertainty is $\Delta T = \frac{\Delta t_{50}}{50} = \frac{1}{50} = 0.02\ \text{s}.$

2. Compute relative uncertainties
Relative uncertainty in length: $\frac{\Delta L}{L} = \frac{0.002}{0.20} = 0.01 = 1\%.$ Relative uncertainty in period: $\frac{\Delta T}{T} = \frac{0.02}{0.80} = 0.025 = 2.5\%.$

3. Propagate uncertainties to $g$
From the formula $g = \frac{4\pi^2 L}{T^2}$, the relative uncertainty in $g$ is $\frac{\Delta g}{g} = \frac{\Delta L}{L} + 2\,\frac{\Delta T}{T} = 0.01 + 2\times 0.025 = 0.01 + 0.05 = 0.06.$

4. Convert to percentage accuracy
Percentage accuracy in $g$ is $100 \times \frac{\Delta g}{g} = 100 \times 0.06 = 6\%.$ Hence $N = 6$, which corresponds to option A.

1. Unit mismatch: Students often forget to convert all lengths to the same unit (meters or centimeters). Here $20\ \text{cm}$ and $2\ \text{mm}$ must both be in meters or both in centimeters before computing $\Delta L/L$. 2. Period uncertainty: The resolution of $1\ \text{s}$ applies to the total time for 50 oscillations, not to one oscillation. Divide by 50 to get $\Delta T$. 3. Error propagation factor: Because $T$ appears squared in the formula for $g$, its relative uncertainty is multiplied by 2, not 1. 4. Rounding too early: Keep intermediate values (like $0.025$) in decimal form until the final percentage to avoid rounding errors.