First slide

Different types of Forces

Question

Three identical rigid circular cylinders A, B and C are arranged on smooth inclined surfaces as shown in figure. The least value of $θ$ that prevent the arrangement from collapse is

Moderate

A

$\tan^{- 1} (\frac{1}{2})$
B

$\tan^{- 1} (\frac{1}{2 \sqrt{3}})$
C

$\tan^{- 1} (\frac{1}{3 \sqrt{3}})$
D

$\tan^{- 1} (\frac{1}{4 \sqrt{3}})$

Solution

W = weight of each sphere
$N_{2}$ = normal reaction between,4 and inclined plane
$N_{1}$ = normal reaction between A and C = normal reaction between B and C
Free body diagram of C:
Resolving vertically $2 N_{1} \cos 30^{0} = W$
or $N_{1} = \frac{W}{\sqrt{3}} - - - - - (1)$
When the arrangement is on the point of collapsing, the reaction between A and B is zero.
Free body diagram of A:
Resolving horizontally and vertically
$N_{2} \sin θ = N_{1} \sin 30^{0} or N_{2} \sin θ = \frac{W}{2 \sqrt{3}} - - (2)$
$N_{2} \cos θ = W + N_{1} \cos 30^{0} = \frac{3 W}{2} - - - (3)$
Dividing (2) by (3) we get
$\tan θ = \frac{1}{3 \sqrt{3}} or θ = \tan^{- 1} (\frac{1}{3 \sqrt{3}})$

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