Question

A particular physical quantity is given by \(X=2\pi \sqrt{\frac{a}{b}}\) and the measured value of 'a' is \(40\) unit with \(0.01\) unit accuracy. The value of 'b' is \(\frac{98}{10}\) unit with \(1\) unit accuracy. The % error in \(X\) is nearly

• A. \(4\%\)
• B. \(5\%\)
• C. \(6\%\)
• D. \(7\%\)

Hint:

For a square root of a ratio, the fractional error is half the sum of the fractional errors of a and b.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
\(X=2\pi\sqrt{\dfrac{a}{b}}\) has powers \(\dfrac{1}{2}\) of a and \(-\dfrac{1}{2}\) of b.
Errors always add up in the maximum error.

Step 2: Error formula
\(\dfrac{\Delta X}{X}=\dfrac{1}{2}\left(\dfrac{\Delta a}{a}+\dfrac{\Delta b}{b}\right)\)

Step 3: Calculation
\(\dfrac{\Delta a}{a}=\dfrac{0.01}{40}=0.00025\)
\(\dfrac{\Delta b}{b}=\dfrac{1}{9.8}\approx 0.102\)
\(\dfrac{\Delta X}{X}=\dfrac{1}{2}(0.00025+0.102)\approx 0.051\)

Step 4: Percentage error
\(\dfrac{\Delta X}{X}\times 100\approx 5\%\)

Final Answer:
The percentage error in X is nearly 5 percent. \[ \boxed{\text{(B) }5\%} \]