Question

In an experiment the values of two spring constants were measured as \(k_1 = (10 \pm 0.2)\) N/m and \(k_2 = (20 \pm 0.3)\) N/m. If these springs are connected in parallel, then the percentage error in equivalent spring constant is:

• A. 1.33\%
• B. 1.67\%
• C. 2.33\%
• D. 2.67\%

Hint:

It's crucial to remember the rules for error propagation.
For addition/subtraction, \textbf{absolute} errors add.
For multiplication/division, \textbf{relative} (or percentage) errors add.
Knowing this distinction is key to solving error analysis problems correctly.

Answer 1

The Correct Option is B

To find the percentage error in the equivalent spring constant when the springs are connected in parallel, we need to follow these steps:

1. Identify the values of the spring constants and their uncertainties:
   • k_1 = (10 \pm 0.2) N/m
   • k_2 = (20 \pm 0.3) N/m
2. When springs are connected in parallel, the equivalent spring constant k_{\text{eq}} is given by:
   k_{\text{eq}} = k_1 + k_2
3. Calculate the equivalent spring constant:
   k_{\text{eq}} = 10 + 20 = 30 \, \text{N/m}
4. Determine the absolute error in k_{\text{eq}} using the formula:
   \Delta k_{\text{eq}} = \Delta k_1 + \Delta k_2
   \Delta k_1 = 0.2 \, \text{N/m}
   \Delta k_2 = 0.3 \, \text{N/m}
   \Delta k_{\text{eq}} = 0.2 + 0.3 = 0.5 \, \text{N/m}
5. Calculate the percentage error in k_{\text{eq}}:
   \text{Percentage Error} = \left( \frac{\Delta k_{\text{eq}}}{k_{\text{eq}}} \right) \times 100\%
   \text{Percentage Error} = \left( \frac{0.5}{30} \right) \times 100\% = 1.67\%

The calculated percentage error in the equivalent spring constant is 1.67%. Therefore, the correct answer is 1.67%.