A sand dropper drops sand of mass m(t) on a conveyer belt at a rate proportional to the square root of speed (v) of the belt, i.e., dm/dt ∝ √(v). If P is the power delivered to run the belt at constant speed then which of the following relationship is true? (a) P ∝ √(v) (b) P ∝ v (c) P² ∝ v³ (d) P² ∝ v⁵

Answer: (d). Belt force F = vdm/dt, so P = dm/dtv². dm/dt ∝ v1/2 ⇒ P ∝ v5/2 P² ∝ v⁵ So the answer is (d).

A sand dropper drops sand of mass m(t) on a conveyer belt at a rate… | JEE Main 2025 (29 Jan, Shift 2)

Q10JEE Main 2025 · 29 JAN · SHIFT 2Difficult

Power

A sand dropper drops sand of mass m(t) on a conveyer belt at a rate proportional to the square root of speed (v) of the belt, i.e., $\frac{dm}{dt} \propto \sqrt{v}$ . If P is the power delivered to run the belt at constant speed then which of the following relationship is true?

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Answer: (d)

Belt force $F = v\frac{dm}{dt}$ , so $P = \frac{dm}{dt}v^2$ .
$\frac{dm}{dt} \propto v^{1/2} \Rightarrow P \propto v^{5/2}$
$P^2 \propto v^5$
So the answer is (d).

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