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Chapter 2: Problem 152

A 5 -gal, cylindrical open container with a bottom area of 120 in. \(^{2}\) is filled with glycerin and rests on the floor of an elevator. (a) Determine the fluid pressure at the bottom of the container when the elevator has an upward acceleration of \(3 \mathrm{ft} / \mathrm{s}^{2}\). (b) What resultsnt force does the container exert on the floor of the elevator during this acceleration? The weight of the container is negligible. (Note: 1 gal \(=231\) in. \(^{3}\) )

Short Answer

Expert verified
First, calculate the height of the fluid, then compute the fluid pressure when the elevator is accelerating upwards. Finally, using this pressure, determine the resultant force exerted on the elevator floor.

Step by step solution

01

Calculate the height of the fluid

To calculate the height of the fluid, we first convert the fluid volume from gallons to cubic inches using the given conversion factor \(1 gal = 231 cu. in.\). Then use the given area of the bottom of the container to determine the height. This can be done using the formula for the volume of a cylinder \(V = A \cdot h\), where \(V\) is the volume, \(A\) is the base area and \(h\) is the height.
02

Compute the fluid pressure

The fluid pressure at the base when the elevator is stationary can be given by \(P1 = \rho * g * h\), where \(\rho\) is the density of fluid, \(g\) is acceleration due to gravity and \(h\) is the height of the fluid column. While the elevator is accelerating upwards, it will add to the effective gravitational pull. The total fluid pressure \(P\) at the base can then be given by \(P = \rho * (g + a) * h\), where \(a\) is the acceleration of the elevator.
03

Determine the resultant force on the elevator floor

The resultant force exerted on the floor of elevator is due to the pressure at the bottom of the glycerin column and can be calculated by \(F = P * A\), where \(F\) is the resultant force, \(P\) is the fluid pressure and \(A\) is the area of the bottom of the container.

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