Question

You measure two quantities as \( A = 1.0 \,m \pm 0.2 \,m \), \( B = 2.0 \,m \pm 0.2 \,m \). We should report the correct value for \( \sqrt{AB} \) as:

• A. \( 1.4 \,m \pm 0.4 \,m \)
• B. \( 1.41 \,m \pm 0.15 \,m \)
• C. \( 1.4 \,m \pm 0.3 \,m \)
• D. \( 1.4 \,m \pm 0.2 \,m \)

Hint:

For multiplication or division, relative errors are added. The formula for error propagation in square roots is: \[ \frac{\Delta Y}{Y} = \frac{1}{2} \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right). \]

Answer 1

The Correct Option is D

Step 1: {Given values}
We are given: \[ A = 1.0 \,m \pm 0.2 \,m, \quad B = 2.0 \,m \pm 0.2 \,m \] We define: \[ Y = \sqrt{AB} \] Step 2: {Calculate the value of \( Y \)}
\[ Y = \sqrt{(1.0)(2.0)} \] \[ = \sqrt{2.0} = 1.414 \,m \] Step 3: {Determine the uncertainty in \( Y \)}
The formula for relative error propagation is: \[ \frac{\Delta Y}{Y} = \frac{1}{2} \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right) \] Substituting the given values: \[ \frac{\Delta Y}{1.4} = \frac{1}{2} \left( \frac{0.2}{1.0} + \frac{0.2}{2.0} \right) \] Step 4: {Simplify the expression}
\[ \frac{\Delta Y}{1.4} = \frac{1}{2} \left( 0.2 + 0.1 \right) \] \[ \frac{\Delta Y}{1.4} = \frac{1}{2} \times 0.3 = 0.15 \] \[ \Delta Y = 0.15 \times 1.4 = 0.21 \] Step 5: {Round off to one significant digit}
\[ \Delta Y = 0.2 \,m \] Thus, the final result is: \[ Y = 1.4 \,m \pm 0.2 \,m \] Step 6: {Verify the options}
Comparing with the given options, the correct answer is (D) \( 1.4 \,m \pm 0.2 \,m \).

Answer 2

Given:
- \( A = 1.0 \,m \pm 0.2 \,m \)
- \( B = 2.0 \,m \pm 0.2 \,m \)
We are asked to report the value of \( \sqrt{AB} \) along with its uncertainty.

Step 1: Calculate the central value
\[ AB = 1.0 \times 2.0 = 2.0 \Rightarrow \sqrt{AB} = \sqrt{2.0} \approx 1.41 \approx 1.4 \,m \]

Step 2: Use error propagation for a function of multiple variables
If \( Q = \sqrt{AB} \), then:
\[ \frac{\Delta Q}{Q} = \frac{1}{2} \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right) \]
Substitute values:
- \( \Delta A = 0.2 \), \( A = 1.0 \Rightarrow \frac{\Delta A}{A} = 0.2 \)
- \( \Delta B = 0.2 \), \( B = 2.0 \Rightarrow \frac{\Delta B}{B} = 0.1 \)
So:
\[ \frac{\Delta Q}{Q} = \frac{1}{2}(0.2 + 0.1) = 0.15 \Rightarrow \Delta Q = Q \times 0.15 = 1.4 \times 0.15 = 0.21 \approx 0.2 \,m \]

Final Answer:
The correct value is \( \boxed{1.4 \,m \pm 0.2 \,m} \)