JEE PYQ: Units & Measurements - Question ID 426ce853589a (JEE Main 2026)

When two springs of constants $k_1$ and $k_2$ are connected in parallel, their equivalent spring constant $k_{\text{eq}}$ is the sum of the individual constants: $k_{\text{eq}} = k_1 + k_2.$ Each measured spring constant carries an absolute uncertainty $\Delta k_1$ and $\Delta k_2$. The absolute uncertainty in the sum of two independent measurements is the sum of their absolute uncertainties: $\Delta k_{\text{eq}} = \Delta k_1 + \Delta k_2.$ The percentage error (relative error) in $k_{\text{eq}}$ is then $\text{Percentage error} = \frac{\Delta k_{\text{eq}}}{k_{\text{eq}}} \times 100\%.$

1. Compute the equivalent spring constant
Given $k_1 = 10\;\mathrm{N/m},\quad k_2 = 20\;\mathrm{N/m},$ we find $k_{\text{eq}} = k_1 + k_2 = 10 + 20 = 30\;\mathrm{N/m}.$

2. Compute the absolute uncertainty in $k_{\text{eq}}$
The uncertainties are $\Delta k_1 = 0.2\;\mathrm{N/m},\quad \Delta k_2 = 0.3\;\mathrm{N/m}.$ Hence $\Delta k_{\text{eq}} = \Delta k_1 + \Delta k_2 = 0.2 + 0.3 = 0.5\;\mathrm{N/m}.$

3. Compute the percentage error
Using the formula for percentage error, $\text{Percentage error} = \frac{\Delta k_{\text{eq}}}{k_{\text{eq}}} \times 100\% = \frac{0.5}{30} \times 100\% = \frac{5}{3}\% \approx 1.67\%.$

4. Match with the given options
The calculated percentage error is $1.67\%$, which corresponds to option C.

1. Mistaking series for parallel: Some students confuse the formula for springs in series ($1/k_{\text{eq}} = 1/k_1 + 1/k_2$) with the parallel case. Always remember that in parallel the constants add directly. 2. Incorrect uncertainty propagation: A frequent error is to add uncertainties in quadrature (i.e.\ $\sqrt{(\Delta k_1)^2 + (\Delta k_2)^2}$) instead of simply summing them. For sums of independent measurements, absolute uncertainties add. 3. Rounding too early: Keep intermediate results (like $0.5/30$) in fractional form to avoid rounding errors before the final percentage.