Fresh water issues from the nozzle with a velocity of 30 m / s at the rate of 0.05 m^3 / s and i

step 1 So here on this problem we have to calculate the force required to hold the vein in place and we

Fresh water issues from the nozzle with a velocity of 30 m /...

Fresh water issues from the nozzle with a velocity of $30 \mathrm{m} / \mathrm{s}$ at the rate of $0.05 \mathrm{m}^{3} / \mathrm{s}$ and is split into two equal streams by the fixed vane and deflected through $60^{\circ}$ as shown. Calculate the force $F$ required to hold the vane in place. The density of water is $1000 \mathrm{kg} / \mathrm{m}^{3}$.

Key Concepts

Conservation of Momentum

In fluid dynamics, the principle of conservation of momentum states that the net force acting on a control volume is equal to the rate of change of momentum of the fluid within that volume. This principle is essential when analyzing forces in systems where the fluid’s velocity direction or magnitude changes, such as when a fluid stream is deflected by a vane.

Mass Flow Rate and Volumetric Flow Rate

The mass flow rate is the amount of mass passing through a given cross-section per unit time, calculated by multiplying the fluid’s density by its volumetric flow rate. This relationship is fundamental in converting volume-based flow measurements into a mass-based framework, which is critical when determining the momentum of the fluid.

Control Volume Analysis

Control volume analysis involves selecting a fixed region in space over which the mass and momentum balances are applied. By examining the net inflow and outflow of momentum, one can identify the forces necessary to resume equilibrium in systems where the flow is redirected, such as in the case of deflected streams.

Fluid Deflection and Resultant Forces

When a fluid stream is deflected, its momentum vector changes direction, resulting in a net change in momentum that must be balanced by an external force. This concept is applied to determine the force required to hold a vane in place, ensuring that the force provided counteracts the reactive momentum change from the deflected fluid.

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