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

Step 1: Use the formula for equivalent spring constant.
The equivalent spring constant \( K_{\text{eq}} \) for two springs connected in parallel is given by: \[ K_{\text{eq}} = k_1 + k_2 \] Substitute the values of \( k_1 \) and \( k_2 \): \[ K_{\text{eq}} = 20 + 30 = 50 \, \text{N/m} \] Step 2: Calculate the total error.
The error in the spring constant is the sum of the individual errors: \[ \Delta K = \Delta k_1 + \Delta k_2 = 0.2 + 0.3 = 0.5 \, \text{N/m} \] Step 3: Calculate the percentage error.
The percentage error is given by: \[ \text{Percentage error} = \frac{\Delta K}{K_{\text{eq}}} \times 100 = \frac{0.5}{50} \times 100 = 1% \] Step 4: Conclusion.
The percentage error in the equivalent spring constant is 1%.