Question

Two springs have spring constants
$k_{1} = (40 \pm 0.4)$ N/m and
$k_{2} = (60 \pm 0.6)$ N/m.
If they are connected in parallel, the percentage error in the equivalent spring constant is:

• A. 0.5%
• B. 1%
• C. 0.75%
• D. 1.5%

Hint:

Remember the rules for error propagation: For addition or subtraction ($Z = A \pm B$), absolute errors add ($\Delta Z = \Delta A + \Delta B$). For multiplication or division ($Z = A \cdot B$ or $Z = A/B$), relative (or percentage) errors add ($\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$). For powers ($Z = A^n$), the relative error is multiplied by the power ($\frac{\Delta Z}{Z} = n \frac{\Delta A}{A}$).

Answer 1

The Correct Option is B

Step 1: Use relative (fractional) error approach

Instead of adding absolute errors directly, we first find the fractional (relative) error of each spring constant.

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Step 2: Calculate fractional errors of individual springs

For the first spring:

Fractional error in k₁ = Δk₁ / k₁ = 0.4 / 40 = 0.01

For the second spring:

Fractional error in k₂ = Δk₂ / k₂ = 0.6 / 60 = 0.01

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Step 3: Find equivalent spring constant

When springs are connected in parallel:

k_eq = k₁ + k₂ = 40 + 60 = 100 N/m

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Step 4: Apply weighted fractional error rule

For addition, the absolute error depends on how much each quantity contributes to the total value.

Since both spring constants have the same fractional error (0.01), the equivalent spring constant will also have the same fractional error.

Fractional error in k_eq = 0.01

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Step 5: Convert to percentage error

Percentage error = 0.01 × 100

Percentage error = 1%

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Final Answer:

The percentage error in the equivalent spring constant is
1%