Question

A tiny metallic rectangular sheet has length and breadth of 5 mm and 2.5 mm, respectively. Using a specially designed screw gauge which has pitch of 0.75 mm and 15 divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum fractional error will be \( \frac{x}{100} \), where \( x \) is:

Hint:

When dealing with measurements involving areas, remember that errors in both dimensions contribute to the total error.

Answer 1

Correct Answer: 3

Step 1: Input Parameters

• Sheet Length, \( L = 5 \) mm
• Sheet Breadth, \( B = 2.5 \) mm
• Screw Gauge Pitch = 0.75 mm
• Circular Scale Divisions = 15
• Screw Gauge Least Count = \( \frac{\text{Pitch}}{\text{Number of divisions}} = \frac{0.75}{15} = 0.05 \) mm

Step 2: Error Determination

The absolute error in screw gauge measurements is equal to its least count, which is \( 0.05 \) mm.

Fractional error in length measurement: \[ \frac{\Delta L}{L} = \frac{0.05}{5} = 0.01 \]

Fractional error in breadth measurement: \[ \frac{\Delta B}{B} = \frac{0.05}{2.5} = 0.02 \]

Step 3: Maximum Fractional Error in Area Calculation

Area, \( A = L \times B \)

Maximum fractional error in area is calculated as: \[ \left( \frac{\Delta A}{A} \right)_{\text{max}} = \left( \frac{\Delta L}{L} + \frac{\Delta B}{B} \right) \] \[ = 0.01 + 0.02 = 0.03 \]

Step 4: Conversion to Specified Format

The fractional error is to be expressed in the form \( \frac{x}{100} \). \[ 0.03 = \frac{x}{100} \] Solving for x: \[ x = 3 \]

Final Result:

\[ \boldsymbol{x = 3} \]