In an operational design test of the equipment of the fire...

In an operational design test of the equipment of the fire truck, the water cannon is delivering fresh water through its 2 -in.-diameter nozzle at the rate of 1400 gal/min at the $20^{\circ}$ angle. Calculate the total friction force $F$ exerted by the pavement on the tires of the truck, which remains in a fixed position with its brakes locked. (There are 231 in. $^{3}$ in 1 gal.)

Key Concepts

Conservation of Momentum

This principle states that the total momentum of an isolated system remains constant if no external forces act on it. In the context of fluid expelled from a nozzle, the rate of momentum change of the moving water must be balanced by an equal and opposite force acting on the system—in this case, the truck. This balance of forces underpins the calculation of the reaction force due to the water flow.

Mass Flow Rate and Volumetric to Mass Conversion

When dealing with fluid flow, it is essential to convert a given volumetric flow rate into a mass flow rate by accounting for the density of the fluid. For water, this involves converting units such as gallons per minute to cubic inches or cubic feet, then using the density of water. This conversion is critical for quantifying the momentum transfer rate in the system.

Momentum Flux of the Fluid

Momentum flux is the rate at which momentum is transferred per unit time by the flowing fluid. It is calculated by multiplying the mass flow rate by the velocity of the fluid. This concept is vital as it directly relates to the force exerted by the fluid, based on the change in momentum as it exits the nozzle at a specified angle.

Newton's Third Law and Reaction Forces

Newton’s Third Law states that for every action, there is an equal and opposite reaction. In this scenario, the momentum imparted to the water as it is discharged produces an equal and opposite force on the fire truck. This reaction force must be balanced by the friction force provided by the pavement to keep the truck stationary.

Unit Conversions in Fluid Dynamics

Ensuring consistency in units across all calculations is essential in fluid dynamics problems. This includes converting volumetric flow rates from gallons per minute to cubic inches (or cubic feet), and then using the density of water to obtain the mass flow rate. Proper unit conversion is a fundamental step that ensures the accuracy of the momentum and force calculations.

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