A U-tube contains water in the right arm, and another liquid...

Name: Problem 66, Chapter 11: Fundamentals of Thermal-Fluid Sciences
Uploaded: 2021-11-04T17:08:14-08:00
Duration: 5 min 19 s
Channel: Tanner Manwaring
Description: A U-tube contains water in the right arm, and another liquid in the left arm. It is observed that when the U-tube rotates at 30 rpm about an axis that is 15 cm from the right arm and 5 cm from the left arm, the liquid levels in both arms become the same. Determine the density of the fluid in the left arm. figure cant copy

A U-tube contains water in the right arm, and another liquid in the left arm. It is observed that when the U-tube rotates at 30 rpm about an axis that is 15 cm from the right arm and 5 cm from the left arm, the liquid levels in both arms become the same. Determine the density of the fluid in the left arm. figure cant copy

Key Concepts

Fluid Density Determination through Pressure Balance

The density of a fluid directly influences its pressure response to acceleration fields. In practical applications like manometry involving rotating systems, differences in density between fluids lead to different heights for the same pressure difference. By establishing a balance in pressures that accounts for both gravitational and centrifugal contributions, one can solve for an unknown density, making density a crucial parameter in the analysis of fluid statics under rotation.

Pressure Distribution in Rotating Fluids

In a rotating fluid, the pressure at any point is influenced by both gravitational acceleration and the additional radial (centrifugal) acceleration. This results in a modified pressure gradient compared to a non-rotating system. The pressure variation must be recalculated by including the influence of rotation, so that the net force, which is the sum of gravitational and centrifugal contributions, is balanced by the gradient of the pressure.

Hydrostatic Equilibrium in Fluids

Hydrostatic equilibrium refers to the balance between pressure gradients within a fluid and the forces acting on it, such as gravity and, in rotating systems, centrifugal effects. In a fluid at rest or steady motion, the pressure gradient at any point counters the local effective acceleration lead by these forces, ensuring that the fluid does not spontaneously move and that its surfaces assume an equilibrium shape.

Centrifugal Force in Rotating Systems

When a system rotates, objects within it experience an apparent outward force known as the centrifugal force. This is not a true force but a result of being in a non-inertial (accelerating) frame. In fluids, this centrifugal force alters the effective acceleration acting on each fluid element, influencing the distribution of pressure and, consequently, the fluid’s equilibrium configuration.

Transcript

00:03 Okay, for this question wants us to find how much water flows out the right arm when a liquid of density 0 .8 grams per centimeters cubed is gradually added to the left arm until the height is 8 centimeters and it doesn't mix with the water so we need to know what volume of water is pushed out by that density of that liquid so what we can use is the energy of is the conservation of energy density and so we know that however much of the liquid is added however much that weighs the same weight of water will exit so if we have our equation we know that the equation for energy density is equal to row gravity times height so what we'll do is we'll have row of the liquid times gravity times the height of the liquid will be equal to row of water times gravity times the height of the liquid will be equal to row of water times the height of the liquid or not the height of the liquid the height of the water exiting so let's plug some numbers in so our density of the liquid is equal to 0 .8 grams per centimeters squared or cubed but we need to get that into standard units so that is equal to 800 kilograms per meter squared meters cubed so that's our density of the liquid.

01:48 Our density of water is equal to 1 ,000 kilograms per meters cubed.

01:58 So let's plug these numbers in.

02:01 So we can cancel out gravity.

02:08 And so our height of the water, which we're looking for, is equal to row of the liquid times the height of the liquid.

02:18 So divided and all over row of water.

02:38 So if we enter those things into our calculator, we'll have 800 times 0 .08 meters, because it's in centimeters, so we'll change it to meters divided by 1 ,000.

03:01 And that answer comes out to be 0 .064 meters for the height.

03:12 So that's 6 .4 meters for the height.

03:13 So that's 6 .4...

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