Physics - NLM Question with Solution | TestHub

All puleys are smooth so tension in string is same. At all points. For horizontal equilibrium at ${\mathrm{P}}_1$ $\mathrm{T}\sin 60^{\circ }=\mathrm{T}\sin \varphi$

$$\Rightarrow \varphi =60^{\circ }$$

For vertical equilibrium at ${\mathrm{P}}_1$ $2\mathrm{T}\cos 60^{\circ }=20$

$$\mathrm{T}=20\mathrm{N}$$

For vertical equilibrium at ${\mathrm{P}}_3$

$$2\mathrm{T}\cos \theta =10$$

$$2\times 20\times \cos \theta =10$$

$$\Rightarrow \cos \theta =\frac{1}{4}\Rightarrow \sin \theta =\frac{\sqrt{15}}{4}$$

⇒ length of string $=2\mathrm{cosec}60^{\circ }+2\mathrm{cosec}\theta =2\times \frac{2}{\sqrt{3}}+2\times \frac{4}{\sqrt{15}}=\frac{4}{\sqrt{3}}\left(1+\frac{2}{\sqrt{5}}\right)$