Understanding Force Formula: ρ AV1 (V1 - V2 cos θ) vs ρ A(V1 - V2 cos θ)

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SUMMARY

The force formula for fluid dynamics is expressed as Fx = ρ AV1 (V1 - V2 cos θ), where ρ represents fluid density, A is the cross-sectional area, V1 is the initial velocity, and V2 is the velocity after a bend. This formula accounts for the rate of flow, Q, which is defined as Q = A1V1, ensuring that the momentum change is accurately represented according to Newton's second law. The distinction between the two formulas arises from the need to incorporate the varying velocities across different cross-sections of the fluid flow.

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hotjohn
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  1. why the Fx is given by the formula = ρ AV1 ( V1 - V2 cos θ ) , why not = ρ A( V1 - V2 cos θ ) ?
 

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The force is given by ρQ(v1 - v2cosθ) where Q is the rate of flow. You can understand this by Newton's second law by observing the rate of change of momentum with time. Note that this rate of flow will remain constant throughout the cross-section (which implies that the cross-section after the bend is different than that before the bend because the fluid has different velocities in the two regions). Thus it suffices to calculate the flow rate (Av) in the upper part where the cross-section and velocity is known, Q = A1v1.
Thus you have your desired result.
 
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