One end of a horizontal uniform beam of weight and length is hinged on a vertical wall at point and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point , at a height above the hinge at point . [2021]

Question

One end of a horizontal uniform beam of weight $W$ and length $L$ is hinged on a vertical wall at point $O$ and its other end is supported by a light inextensible rope. The other end of the rope is fixed at point $Q$ , at a height $L$ above the hinge at point $O$ . A block of weight $α W$ is attached at the point $P$ of the beam, as shown in the figure (not to scale). The rope can sustain a maximum tension of $(2 \sqrt{2}) W$ . Which of the following statement(s) is(are) correct [2021]

Smart Achievers · Accepted Answer

(1, 2, 4)

Since $O Q = O P$ $∴ ∠ P = ∠ Q = 45 °$

At equilibrium, about point $O$ ,

$R_{y} + \frac{T}{\sqrt{2}} = W + α W$ ...(i)

and $R_{x} = \frac{T}{\sqrt{2}}$ ...(ii)

Torque about point $' O'$ is zero,

So, $W \frac{L}{2} + α W L = \frac{T}{\sqrt{2}} L$ $∴$ $T = \sqrt{2} (\frac{W}{2} + α W)$ ...(iii)

$∴$ $R_{x} = \frac{T}{\sqrt{2}} = (\frac{W}{2} + α W)$

Therefore for $α = 0.5$ ,

$R_{x} = \frac{W}{2} + α W = \frac{W}{2} + 0.5 W or R_{x} = W$

i.e., the horizontal component of reaction force at $O$ ,

$R_{x} = W for α = 0.5$

Now torque about point P,

$T_{y} L = W \cdot \frac{L}{2} \Rightarrow R_{y} = \frac{W}{2}$

The vertical component of reaction force at O does not depend on $α$ .

As per question, rope can sustain a maximum tension of $2 \sqrt{2} W$ ,

$∴$ $2 \sqrt{2} W = \sqrt{2} (\frac{W}{2} + α W)$

$\Rightarrow 2 = \frac{1}{2} + α$

$∴$ $α = \frac{3}{2}$

Solution

1 The vertical component of reaction force at $O$ does not depend on $α$

2 The horizontal component of reaction force at $O$ is equal to $W$ for $α = 0.5$

3 The tension in the rope is $2 W$ for $α = 0.5$

4 The rope breaks if $α > 1.5$

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Solution

1 The vertical component of reaction force at O does not depend on α

2 The horizontal component of reaction force at O is equal to W for α=0.5

3 The tension in the rope is 2W for α=0.5

4 The rope breaks if α>1.5

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1 The vertical component of reaction force at $O$ does not depend on $α$

2 The horizontal component of reaction force at $O$ is equal to $W$ for $α = 0.5$

3 The tension in the rope is $2 W$ for $α = 0.5$

4 The rope breaks if $α > 1.5$