Question

There are two springs of spring constants \( k_1 = (20 \pm 0.2) \, \text{N/m \) and \( k_2 = (30 \pm 0.3) \, \text{N/m} \). If they are connected in parallel, then the percentage error in the equivalent spring constant of the combination is ........... %.}

Hint:

When combining springs in parallel, the total error is the sum of the individual errors, and the percentage error is calculated as the error divided by the total value.

Answer 1

Correct Answer: 1

In a parallel configuration, the equivalent spring constant \( k_{\text{eq}} \) is given by the sum of the individual spring constants: \( k_{\text{eq}} = k_1 + k_2 \). Here, \( k_1 = 20 \pm 0.2 \, \text{N/m} \) and \( k_2 = 30 \pm 0.3 \, \text{N/m} \). Thus, \( k_{\text{eq}} = 20 + 30 = 50 \, \text{N/m} \).
To find the percentage error in \( k_{\text{eq}} \), use the formula for percentage error: \(\left(\frac{\Delta k_{\text{eq}}}{k_{\text{eq}}}\right) \times 100\%\). Assuming independent measurements, errors add linearly: \(\Delta k_{\text{eq}} = \Delta k_1 + \Delta k_2 = 0.2 + 0.3 = 0.5 \, \text{N/m}\).
Hence, the percentage error is \(\left(\frac{0.5}{50}\right) \times 100\% = 1\%\).
This matches the expected range of 1 to 1. Therefore, the percentage error in the equivalent spring constant is precisely 1%.