GATE 2023 CE (CE2) – Question 53

Engineering Mechanics · System of forces, free-body diagrams, equilibrium equations · 2 marks · Numerical answer

A beam is subjected to a system of coplanar forces as shown in the figure. The magnitude of vertical reaction at Support P is ______ N (round off to one decimal place).

simply supported beam P–Q of length 6 m; a 500 N force acts downward/right at 60° to the beam at x = 2 m, a 200 N force acts upward at x = 3.5 m, and a 100 N rightward force acts 0.5 m above the beam at x = 4.5 m.

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Correct answer: 196.02 to 197.98

Explanation

Take moments about support Q to find the vertical reaction at P. Counter-clockwise is positive.

**Loads (from the figure).**
- 500 N at 60° to the beam at $x=2$ m. Its vertical component is $500\sin60^\circ=433.0$ N downward, and the horizontal component acts along the beam, so it has no moment about Q.
- 200 N upward at $x=3.5$ m.
- 100 N horizontal at 0.5 m above the beam at $x=4.5$ m, which gives a moment about Q because it acts off the beam axis.

**Moments about Q** (the beam length is 6 m):
- Vertical 433.0 N at 4 m from Q: $+433.0\times4=+1732$ N m
- 200 N upward at 2.5 m from Q: $-200\times2.5=-500$ N m
- 100 N horizontal at height 0.5 m: $-100\times0.5=-50$ N m
- $R_P$ at 6 m from Q: $-6R_P$

$$1732-500-50-6R_P=0\;\Rightarrow\;R_P=\frac{1182}{6}=\mathbf{197.0\ N}\ (\text{upward}).$$