First slide

Equilibrium of a particle

Question

Three identical rigid circular cylinders A, B and C are placed on smooth inclined plane shown in the figure. The minimum value of tan $θ$ that prevent the arrangement from collapse is $\frac{1}{3 \sqrt{n}}$ . The value of n is

Moderate

Solution

From free body diagram of cylinder A, $2 N \cos 30^{\circ} = m g$
$∴ N = \frac{m g}{\sqrt{3}}$
From free body diagram of cylinder B,
$\begin{array}{l} N_{0} \cos θ = mg + Ncos 30^{\circ} \\ N_{0} \sin θ = Nsin 30^{\circ} \\ \tan θ = \frac{N \sin 30^{\circ}}{m g + N \cos 30^{\circ}} \\ \Rightarrow \tan θ = \frac{\frac{m g}{\sqrt{3}} \sin 30^{\circ}}{m g + \frac{m g}{\sqrt{3}} \cos 30^{\circ}} = \frac{1}{3 \sqrt{3}} \end{array}$

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