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A circular ring of radius $' r'$ metre is suspended from a point $' h'$ metre vertically above the centre of ring by four strings of equal lengths attached at equal intervals to the circumference of the ring. If $' α'$ be the angle between two consecutive string, then $\cos α$ is

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$\frac{h^{2}}{h^{2} + r^{2}}$

$\frac{r^{2}}{h^{2} + r^{2}}$

$\frac{2 h^{2}}{h^{2} + r^{2}}$

$\frac{2 r^{2}}{h^{2} + r^{2}}$

answer is A.

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Detailed Solution

Let $' O'$ be the centre of ring $' P'$ be the point of suspension $P A, P B$ are two consecutive string, then $∠ A O B = \frac{π}{2}$

$O A = O B = r, O P = h$

In $Δ A O B$ ,

$A B^{2} = O A^{2} + O B^{2} = r^{2} + r^{2} = 2 r^{2}$

$\Rightarrow A B = r \sqrt{2}$

In $Δ O P B$

$P B^{2} = O P^{2} + O B^{2} = h^{2} + r^{2}$

$P A = P B = \sqrt{h^{2} + r^{2}}$

If $∠ A P B = α$

In $Δ A P B$ ,

$\cos α = \frac{P A^{2} + P B^{2} - A B^{2}}{2 P A \cdot P B}$

$= \frac{2 (h^{2} + r^{2}) - 2 r^{2}}{2 (h^{2} + r^{2})}$

$= \frac{h^{2}}{h^{2} + r^{2}}$

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