JEE PYQ: Units & Measurements - Question ID cb8723255ab8 (JEE Main 2024)

Core Formula & Concept:

In simple harmonic motion (SHM) of a mass-spring system, the time period $T$ of oscillation is given by: $T = 2\pi \sqrt{\frac{m}{k}}$ where

• $m$ is the mass attached to the spring,
• $k$ is the spring constant.

To find the spring constant experimentally, we rearrange the formula: $k = \frac{4\pi^2 m}{T^2}$ This expression shows that $k$ depends on $m$ and $T^2$. Any errors in measuring $m$ or $T$ will propagate to the calculated value of $k$.

Step-by-Step Derivation:

Step 1: Express the relative error in $k$
Since $k = \frac{4\pi^2 m}{T^2}$, we take the natural logarithm on both sides to find the relative error: $\ln k = \ln(4\pi^2) + \ln m - 2 \ln T$ Differentiating both sides with respect to the measured quantities: $\frac{dk}{k} = \frac{dm}{m} - 2 \frac{dT}{T}$ The relative error in $k$ is: $\frac{\Delta k}{k} = \frac{\Delta m}{m} - 2 \frac{\Delta T}{T}$

Step 2: Substitute the given percentage errors
The student commits:

• $1\%$ negative error in mass, so $\frac{\Delta m}{m} = -1\%$,
• $2\%$ positive error in time, so $\frac{\Delta T}{T} = +2\%$.

Substituting these values: $\frac{\Delta k}{k} = (-1\%) - 2(+2\%) = -1\% - 4\% = -5\%$ The negative sign indicates the direction of error, but the percentage error is the magnitude, so $5\%$.

Step 3: Conclusion
The percentage error in determining $k$ is $5\%$. Hence, the correct option is A: 5%.

Common Traps & Exam Tip:

• Sign Confusion: Students often forget that a negative error in mass ($\Delta m/m = -1\%$) and a positive error in time ($\Delta T/T = +2\%$) combine differently. The negative sign in the formula $\frac{\Delta k}{k} = \frac{\Delta m}{m} - 2 \frac{\Delta T}{T}$ is crucial.
• Squaring the Time Error: Since $k \propto T^{-2}$, the error in $T$ is multiplied by 2, not 1. Many students mistakenly treat it as a linear dependence.
• Magnitude vs. Direction: The question asks for the percentage error, not the direction. Even if the result is negative, the magnitude is what matters.

Always write down the error propagation formula explicitly to avoid such mistakes.