Question

A rod of length $2 \,cm$ makes an angle $\frac{2 \pi}{3}$ 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 $\alpha rad$. The value of $\alpha$ is $\frac{\pi}{n} rad$, where $n$ is ___

Answer 1

$\frac{ h _{ i }}{ h _0}=\frac{ v }{ u } \Rightarrow \frac{-\frac{30 \sqrt{3}}{13}}{\sqrt{3}}=\frac{ v }{-\frac{43}{3}} \Rightarrow v _1=\frac{430}{13} cm$
$ * \frac{1}{ v }-\frac{1}{ u }=\frac{1}{ f } \Rightarrow \frac{1}{ v }=\frac{1}{10}-\frac{3}{40} \Rightarrow v =40 cm $
$ * x =40-\frac{430}{13}=\frac{90}{13} cm$
$ \tan \alpha=\frac{\frac{30 \sqrt{3}}{13}}{\frac{90}{13}}=\frac{1}{\sqrt{3}} $
$\Rightarrow \alpha=30^{\circ}=\frac{\pi}{6}$
$N =6$