A gas having density ρflows with a velocity v along a pipe...

A gas having density $\rho$ flows with a velocity $v$ along a pipe of cross-sectional area $s$ and bent at an angle of $90^{\circ}$ at a point $A$. The force exerted by the gas on the pipe at $A$ is (A) $\frac{\sqrt{2} s v}{\rho}$ (B) $s v^{2} \rho$ (C) $\frac{\sqrt{3} s v^{2} \rho}{2}$ (D) $s v^{2} \rho$

A gas having density $\rho$ flows with a velocity $v$ along a pipe of cross-sectional area $s$ and bent at an angle of $90^{\circ}$ at a point $A$. The force exerted by the gas on the pipe at $A$ is
(A) $\frac{\sqrt{2} s v}{\rho}$
(B) $s v^{2} \rho$
(C) $\frac{\sqrt{3} s v^{2} \rho}{2}$
(D) $s v^{2} \rho$

Key Concepts

Change in Momentum (Impulse)

The change in momentum, or impulse, is a measure of the effect of a net force acting over a period of time. In the context of a fluid making a bend in a pipe, the direction of the fluid’s momentum changes, which according to Newton's second law results in a force being exerted on the pipe. This is critical for understanding the dynamic interactions between the fluid and the pipe structure.

Newton's Second Law in Fluid Dynamics

Newton's second law states that the force on an object is equal to the rate of change of its momentum. In fluid dynamics, applying this law involves calculating how the momentum flux is altered by bends, contractions, or expansions in the flow channel, and then relating this change to the force exerted on the piping system.

Momentum Flux

Momentum flux is the rate at which momentum is transferred through a unit area. In fluid dynamics, this concept is used to quantify how much momentum is carried by the flowing fluid and how changes in the flow (such as a directional change) can result in forces acting on the boundaries of the flow channel.

Mass Flow Rate

The mass flow rate is defined as the mass of the fluid passing through a given cross-sectional area per unit time. It is calculated as the product of the fluid’s density, the cross-sectional area, and the flow velocity. This parameter is essential for determining how much momentum is being transported by the fluid.

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