Physics Solution: Fluid Mechanics Momentum Balance

Physics Solution: Force on U-Tube Assembly

Core Principle: Momentum Flux
The force exerted by a fluid on a pipe bend is equal to the rate of change of momentum of the fluid. For a 180° turn (U-tube), the velocity changes from $+v$ to $-v$, so $\Delta v = 2v$. The force magnitude is $F = \dot{m} \Delta v$.

Step-by-Step Calculation

1. Calculate Force from Left Tube
The water enters with speed $u$ and exits with speed $u$ in the opposite direction.

Mass flow rate: $\dot{m}_L = \rho A u$
Change in velocity: $\Delta v = 2u$
Force exerted on the tube: $F_L = \dot{m}_L \Delta v = (\rho A u)(2u) = 2\rho A u^2$
Direction: Since the water is pushed to the left, the force on the tube is to the Right.

2. Calculate Force from Right Tube
The water enters with speed $v$ and exits with speed $v$.

Mass flow rate: $\dot{m}_R = \rho (A/2) v$
Change in velocity: $\Delta v = 2v$
Force exerted on the tube: $F_R = \dot{m}_R \Delta v = (\rho \frac{A}{2} v)(2v) = \rho A v^2$
Direction: Based on the diagram, this U-tube faces the opposite way, exerting force to the Left.

3. Balance the Forces
The problem states the net force on the assembly is zero. Thus, the magnitudes must be equal: $$F_L = F_R$$ $$2\rho A u^2 = \rho A v^2$$ Dividing both sides by $\rho A$: $$2u^2 = v^2$$ $$v = \sqrt{2u^2} = u\sqrt{2}$$