Question

An object is placed at a distance of 40 cm from the pole of a converging mirror. The image is formed at a distance of 120 cm from the mirror on the same side. If the focal length is measured with a scale where each 1 cm has 20 equal divisions. If the fractional error in the measurement of focal length is \(\frac{1}{10\ k}\).Find k.

Answer 1

Correct Answer: 60

The lens formula is given by:

\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \] 

Where:

• \(u = -40 \ \text{cm} \ \text{(object distance)}\)
• \(v = -120 \ \text{cm} \ \text{(image distance)}\)

 

Substituting the values into the formula:

\[ \frac{1}{-120} + \frac{1}{-40} = \frac{1}{f} \]

Simplify the terms:

\[ \frac{-1}{120} + \frac{-1}{40} = \frac{-1 - 3}{120} = \frac{-4}{120} \]

Thus:

\[ \frac{1}{f} = \frac{-4}{120} = \frac{-1}{30} \]

Taking the reciprocal:

\[ f = -30 \ \text{cm} \]

Calculating Least Count and Error:

The least count of the scale is:

\[ \text{Least Count} = \frac{1}{20} \ \text{cm} \]

The fractional error in the measurement is:

\[ \text{Fractional Error} = \frac{1}{20 \times 30} = \frac{1}{600} \]

Expressing the error as a factor of \(k\):

\[ \frac{1}{10k} = \frac{1}{600} \]

Solving for \(k\):

\[ 10k = 600 \quad \Rightarrow \quad k = 60 \]

Final Answer:

\(k = 60\)