Consider two water tanks filled with water. The first tank...

Consider two water tanks filled with water. The first tank is 8 m high, and is stationary, while the second tank is 2 m high, and is moving upward with an acceleration of $5 \mathrm{~m} / \mathrm{s}^2$. Which tank will have a higher pressure at the bottom?

Key Concepts

Non-Inertial Reference Frames

A non-inertial reference frame is one that is accelerating relative to an inertial frame. In these frames, fictitious or pseudo-forces must be introduced to accurately describe the motion of objects or the behavior of fluids. When considering fluid statics in a non-inertial frame, these pseudo-forces alter the effective gravitational acceleration, thereby affecting calculations such as the pressure distribution within the fluid.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It is determined by the product of the fluid's density, the acceleration due to gravity, and the height (depth) of the fluid column above the point in question. This concept is central in understanding how pressure changes with depth in any fluid, regardless of whether the container is accelerating or stationary.

Effective Gravitational Acceleration

Effective gravitational acceleration refers to the apparent acceleration due to gravity experienced within a system that is accelerating. In a non-inertial reference frame, such as an accelerating container, the effective gravitational force is modified by the acceleration of the system. This means that the standard gravitational acceleration (g) is effectively increased or decreased by the acceleration of the reference frame when calculating quantities like hydrostatic pressure.

Transcript

00:01 So what we want to do is we want to find the total force that's pushing up against the water at the bottom of the tank as well as the pressure so what we're first going to do is find the volume and we're going to do this by using this equation here where we take the height and we multiply it by pi times d squared over four since this expression right here represents the cross -sectional area of the cylinder what we're going to find is that the volume of the first tank is equal to the volume of the first tank is equal to the volume with the second tank.

00:35 And this is, they're both equal to a value of 31 .416 meters cubed.

00:44 So because their volumes are the same, we know that their masses are going to be the same as well.

00:50 Because we're dealing with the same liquid, water in this case, in both tanks.

00:56 So now what we're going to do is we're going to find the force on the bottom of the tank due to gravity.

01:03 So what we're going to do is we're going to take the mass, multiply it by the graph, gravitational acceleration, 9 .1 in this case.

01:11 And we're going to substitute for m density times the volume.

01:17 And when we plug in for the values that we know, we're going to end up getting as a force of 308 .1 kilenoons.

01:29 Now if we want to find the total force, we have to take into account the force due to atmospheric pressure.

01:36 So on the first tank, the total force will be equal to fw what we just found here, plus the atmospheric pressure times the cross -sectional area in the first tank.

01:55 So again, for the area, we use this expression here and plugging in for the values that we know and that we're given.

02:05 What we end up getting is a force of 626 .4 kiloons.

02:17 Now for the second tank, again we're following the same formula but just substituting in for the cross -sectional area of the second tank...

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