A uniform rectangular plate is hanging vertically downward from a hinge that passes along its left edge. By blowing air at $$12m/s$$ over the top of the plate only, it is possible to keep the plate in a horizontal position, as illustrated in part (a) of the drawing. To what value of speed (in$$ $$m/s) should the air be blown so that the plate is kept at $$30$$∘angle with respect to the vertical, as in part (b) of the drawing? (assume$$ the velocity beneath the plate is almost $$zero)

Question

A uniform rectangular plate is hanging vertically downward from a hinge that passes along its left edge. By blowing air at $$12m/s$$ over the top of the plate only, it is possible to keep the plate in a horizontal position, as illustrated in part (a) of the drawing. To what value of speed (in$$ $$m/s) should the air be blown so that the plate is kept at $$30$$∘angle with respect to the vertical, as in part (b) of the drawing? (assume$$ the velocity beneath the plate is almost $$zero)

Infinity Learn · Accepted Answer

Case (I)Due$$ to moving air at the top of the plate, pressure at the top is less than that at the bottom $$(due$$ to Bernoulli’s $$theorem)Δ$$p=12$$ρ$$v2F=$$Δp.$$A(Where$$ A is area of the plate;  F acts at the midpoint of the plate and directed $$upwards)Mg$$ acts as the midpoint of the plate and directed downwards.∴ For equilibrium, balance moments about the hinge. $$12$$ρ$$v2$$.A.$$L2=Mg$$.$$L2$$.....$$.1$$ Case (II)F$$'$$=12$$ρv'$$2A$$∴ For equilibrium, balance moments about the $$hinge.12$$ρv'$$2$$.A.$$L2=Mg$$.$$L2sin30$$∘......$$.2Dividing$$ $$21$$, we get​⇒v'$$=v2=122=62$$ $$ms-1Therefore$$, the correct answer is  $$(B)

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