Question

A rod of length $2cm$ makes an angle $\frac{2\pi }{3}$ rad with the principal axis of a thin convex lens. The lens has a focal length of $10cm$ 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

Answer: 6

$\frac{h_i}{h_0}=\frac{v}{u}\Rightarrow \frac{-\frac{30\sqrt{3}}{13}}{\sqrt{3}}=\frac{v}{-\frac{43}{3}}\Rightarrow v_1=\frac{430}{13}cm$
$\ast \frac{1}{v}-\frac{1}{u}=\frac{1}{f}\Rightarrow \frac{1}{v}=\frac{1}{10}-\frac{3}{40}\Rightarrow v=40cm$
$\ast x=40-\frac{430}{13}=\frac{90}{13}cm$
$\tan \alpha =\frac{\frac{30\sqrt{3}}{13}}{\frac{90}{13}}=\frac{1}{\sqrt{3}}$
$\Rightarrow \alpha =30^{\circ }=\frac{\pi }{6}$
$N=6$