JEE Advanced 2021 Paper 2 Q01 Physics Mechanics Work-Energy-Power Medium

JEE Advanced 2021 Paper 2 · Q01 · Work-Energy-Power

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 $\alpha W$ 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?

[Figure: A horizontal beam OP of length L hinged at O on a vertical wall; a rope from end P goes up to point Q on the wall at height L above O (so OQ = L, OP = L, rope PQ at 45 degrees). A block of weight alpha*W hangs from end P.]

  1. A. The vertical component of reaction force at O does not depend on $\alpha$
  2. B. The horizontal component of reaction force at O is equal to $W$ for $\alpha = 0.5$
  3. C. The tension in the rope is $2W$ for $\alpha = 0.5$
  4. D. The rope breaks if $\alpha > 1.5$
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:A, B, D

Solution

Let T be the tension and $R_1$ (horizontal) and $R_2$ (vertical) be the components of the hinge reaction. The rope makes $45^\circ$ with the beam. Force balance: horizontal $R_1 = T\cos 45^\circ = T/\sqrt{2}$; vertical $R_2 + T\sin 45^\circ = W + \alpha W$, so $R_2 = W(1+\alpha) - T/\sqrt{2}$. Torque about O: $W(L/2) + \alpha W L = T\sin 45^\circ \cdot L$, giving $T = \sqrt{2}\,W(\alpha + 1/2)$. Therefore $R_1 = W(\alpha + 1/2)$ and $R_2 = W/2$, independent of $\alpha$ — (A) correct. For $\alpha = 0.5$: $R_1 = W$ — (B) correct; $T = \sqrt{2}\,W$, not $2W$ — (C) incorrect. Rope breaks when $T > 2\sqrt{2}\,W$, i.e. $\sqrt{2}\,W(\alpha + 0.5) > 2\sqrt{2}\,W \Rightarrow \alpha > 1.5$ — (D) correct.

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