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The force F exerted on an object by a water stream depends on the density of water (rho, in kg/m³), the speed of the stream (v, in m/s), and the cross-sectional area of the stream (A, in m²). Using dimensional analysis, find the expression for F.

1. rho * A * v
2. rho * A * v²
3. rho² * A * v
4. rho * A² * v

Correct answer: rho * A * v²

Solution

Let F = k * rho^a * A^b * v^c. Dimensionally: M L T⁻² = (M L⁻³)^a * (L²)^b * (L T⁻¹)^c = M^a * L^(-3a+2b+c) * T^(-c). Matching: a=1, -c=-2 so c=2, -3+2b+2=1 so b=1. F = k * rho * A * v².

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