A rod of length 2 cm makes an angle with the principal axis of a thin convex lens. The lens has a focal length of 10 cm and is placed at a distance of from the object as shown in the figure. The height of the image is and the angle made by it with resp

Question

A rod of length 2 cm makes an angle $\frac{2 π}{3} rad$ with the principal axis of a thin convex lens. The lens has a focal length of 10 cm and is placed at a distance of $\frac{40}{3} cm$ from the object as shown in the figure. The height of the image is $\frac{30 \sqrt{3}}{13} cm$ and the angle made by it with respect to the principal axis is $α rad$ . The value of $α$ is $\frac{π}{n} rad$ , where $n$ is ________. [2022]

Smart Achievers · Accepted Answer

(6)

By lens formula, $\frac{1}{v} = \frac{1}{f} + \frac{1}{u} \Rightarrow v = \frac{u f}{f + u}$

So, $O A' = \frac{(- \frac{40}{3}) \times 10}{- \frac{40}{3} + 10} = 40 cm$

and, $O B' = \frac{(- \frac{43}{3}) \times 10}{- \frac{43}{3} + 10} = \frac{430}{13} cm$

So, $A' B' = O A' - O B' = 40 - \frac{430}{13} = \frac{90}{13} cm$

Therefore, $\tan α = \frac{\frac{30 \sqrt{3}}{13}}{\frac{90}{13}} = \frac{1}{\sqrt{3}}$ $\Rightarrow α = 30 ° = \frac{π}{6}$

So, $n = 6$

Solution

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Solution

Ans. (6) By lens formula, 1v=1f+1u⇒v=uff+u So, OA'=(-403)×10-403+10=40 cm and, OB'=(-433)×10-433+10=43013cm So, A'B'=OA'-OB'=40-43013=9013 cm Therefore, tanα=303139013=13⇒α=30°=π6 So, n=6

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