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Question: A kite of weight W is flying with its string along a straight line. If the ratios of the resultant a...

A kite of weight W is flying with its string along a straight line. If the ratios of the resultant air pressure R to the tension T in the string and to the weight of the kite are 2\sqrt{2} and (3+1)(\sqrt{3} + 1) respectively, then

A

T=(6+2)WT = (\sqrt{6} + \sqrt{2})W

B

R=(3+1)WR = (\sqrt{3} + 1)W

C

T=12(6−2)WT = \frac{1}{2}(\sqrt{6} - \sqrt{2})W

D

R=(3−1)WR = (\sqrt{3} - 1)W

Answer

R=(3+1)WR = (\sqrt{3} + 1)W

Explanation

Solution

From Lami's theorem,

Rsin⁡(θ+φ)=Tsin⁡(1800−θ)=Wsin⁡(1800−φ)\frac{R}{\sin(\theta + \varphi)} = \frac{T}{\sin(180^{0} - \theta)} = \frac{W}{\sin(180^{0} - \varphi)}

⇒Rsin⁡(θ+φ)=Tsin⁡θ=Wsin⁡φ\Rightarrow \frac{R}{\sin(\theta + \varphi)} = \frac{T}{\sin\theta} = \frac{W}{\sin\varphi} .....(i)

Given, RT=2\frac{R}{T} = \sqrt{2} .....(ii) and RW=3+1\frac{R}{W} = \sqrt{3} + 1 .....(iii)

Dividing (iii) by (ii), we get RWRT=3+12\frac{\frac{R}{W}}{\frac{R}{T}} = \frac{\sqrt{3} + 1}{\sqrt{2}}

⇒TW=3+12⇒T=3+12W=12(6+2)W⇒R=T2=22(3+1)W=(3+1)W\Rightarrow \frac{T}{W} = \frac{\sqrt{3} + 1}{\sqrt{2}} \Rightarrow T = \frac{\sqrt{3} + 1}{\sqrt{2}}W = \frac{1}{2}(\sqrt{6} + \sqrt{2})W \Rightarrow R = T\sqrt{2} = \frac{\sqrt{2}}{\sqrt{2}}(\sqrt{3} + 1)W = (\sqrt{3} + 1)W