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

Step 1: Understanding the Question:
We have two measured values, each with an associated absolute error.
We need to combine these values according to the rule for springs in parallel and then find the percentage error of the resulting quantity.
Step 2: Key Formula or Approach:
1. Springs in Parallel: For springs connected in parallel, the equivalent spring constant \(k_{eq}\) is the sum of the individual constants: \(k_{eq} = k_1 + k_2\).
2. Error Propagation for Addition: When adding quantities, their absolute errors add up. If \(Z = A + B\), then the absolute error in Z is \(\Delta Z = \Delta A + \Delta B\).
3. Percentage Error: The percentage error in a measurement Z is calculated as \(\% \text{Error} = \frac{\Delta Z}{|Z|} \times 100\%\).
Step 3: Detailed Explanation:
The given spring constants and their absolute errors are:
\(k_1 = 10\) N/m and \(\Delta k_1 = 0.2\) N/m.
\(k_2 = 20\) N/m and \(\Delta k_2 = 0.3\) N/m.
First, we calculate the nominal value of the equivalent spring constant for the parallel combination.
\[ k_{eq} = k_1 + k_2 = 10 \text{ N/m} + 20 \text{ N/m} = 30 \text{ N/m}.
\] Next, we calculate the absolute error in the equivalent spring constant. For addition, the absolute errors add.
\[ \Delta k_{eq} = \Delta k_1 + \Delta k_2 = 0.2 \text{ N/m} + 0.3 \text{ N/m} = 0.5 \text{ N/m}.
\] So, the full value of the equivalent spring constant is \(k_{eq} = (30 \pm 0.5)\) N/m.
Finally, we calculate the percentage error in \(k_{eq}\).
\[ \% \text{ error} = \frac{\Delta k_{eq}}{k_{eq}} \times 100\%
\] \[ \% \text{ error} = \frac{0.5}{30} \times 100\% = \frac{50}{30}\% = \frac{5}{3}\%.
\] Converting the fraction to a decimal gives:
\[ \frac{5}{3} \% \approx 1.666... \%.
\] Rounding to two decimal places, the percentage error is 1.67\%.
Step 4: Final Answer:
The percentage error in the equivalent spring constant is 1.67\%.