The pulleys and strings shown in figure are smooth and of negligible mass. For the system to remain in…

The pulleys and strings shown in figure are smooth and of negligible mass. For the system to remain in equilibrium, angle $\theta$ should be
  1. $\quad \cos ^{-1}(1)$
  2. $\quad \cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$
  3. $\cos ^{-1}\left(\frac{1}{2}\right)$
  4. $\quad \cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)$

Solution

From the diagram,
$\mathrm{T}=\mathrm{Mg}$ and, For the system to remain in equilibrium, $\begin{array}{ll} & 2 \mathrm{~T} \cos \theta=\sqrt{2} \cdot \mathrm{Mg} \\ \therefore & 2 \mathrm{~T} \cos \theta=\mathrm{T} \sqrt{2} \\ \therefore & \cos \theta=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}} \\ \therefore & \theta=\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right) \end{array}$

Asked in: MHT CET 2024 (15 May Shift 1)

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