Most of us have an intuitive understanding of Fluid Pressure. We blow air into tires and they become hard with pressure. If we blow up a balloon too much it will pop from the pressure. We may have also seen videos where balloons will expand continuously and pop once reaching the upper atmosphere, or that compressible items will shrink when brought deep beneath the ocean.
The ambient air around us works similarly, constantly pressing on us from all directions at all times. The reason why balloons expand continuously and pop when the atmosphere gets thin is because there is not enough air pressure oustide the walls of the balloon pushing back against the air pressurizing the inside of the balloon.
In some facilities where cleanliness is a critical factor (such as hospitals) the air inside the building is kept slightly pressurized compared to outside, to encourage dust and dirt to blow outside when doors are opened. What if we wanted to create a simple device that we can look at to quickly determine if the pressure inside the building is greater or less than outside the building? It turns out that we could make such a device called a Differential Manometer, using just a bent tube and some fluid.
A U-Bend Differential Manometer.
Here the pressure P_B is greater than P_A, causing h_B to be less than h_A.
Let's imagine we have some cylindrical tube in the shape of a U-bend which is open on both
ends, and inside we pour some liquid (like water) which completely covers the bend. We then
place this tube inside the wall of our building so that one end is inside and the other end
is outside. Initially when the internal Building Pressure P_B and the Atmospheric Pressure
P_A outside the building are equal, the fluid levels in both straight sections of the tube
inside and outside the building are at the same height. However after the building begins to
pressurize itself, P_B will increase while P_A stays constant. Observing the tube, we
would expect the fluid level height in the "inside" section of the tube to decrease, while
the fluid level height in the "outside" section of the tube to increase. But why does that
happen?
First we should recognize that Pressure is a Force applied to an Area. We can
have an intuitive sense of this because it "pushes" on things. In the same manner Weight
can also be understood as a Force, as heavy objects push downwards on whatever is beneath them.
We can calculate this force of weight with the equation F = m * a, where F is the force of
weight, m is the mass of the object, and a is the acceleration (in this case, the acceleration
due to gravity g ~= -9.8 m/s^2).
In our example, lets define some height near the bottom of the manometer (but still within
the constant-radius sections of the straight tubes) as h = 0. For each of the little slices of
fluid in the cross-sectional area of each straight section, let's calculate the total force pushing
downwards on each little area. Each total downward force is comprised of two forces - the force applied by the Ambient
Pressure, and the force applied by the weight of the cylinder of fluid in that tube. Let's recall
that we can calculate the mass of a cylinder given it's density ρ (called "rho"), multiplied
by it's height h, multiplied again by it's cross-sectional area c. However when calculating the
force over the cross-sectional area, c ends up dividing itself out. Therefore, the total downward force
being applied on the cross-sectional area at h=0 on each side:
$$\huge F_A = P_A + \rho g h_A$$ $$\huge F_B = P_B + \rho g h_B$$
However, let's think about this for a moment. When a force is applied to an object then it causes
that object to move, unless there is an equal force in the opposite direction to cancel it out.
But in our example once both pressure's have stabilized, the fluid should not be moving! The
magic here is that due to the bend in our manometer, any force which pushes the fluid downward
on one side of the tube is actually also pushing the fluid in the other tube upwards, and so
the fact that the fluid is stationary implies that F_A = F_B.
Problem Statement
Take a differential manometer like the one described above. Initially both inside and outside
pressures are equal to P_A, and so both fluid levels are equal. The plan is to change the
building's internal air to P_B. As the building's air pressure changes you watch the fluid level
inside the building change as well. How far above or below will the fluid level change from the
initial amount after the inside pressure has reached P_B, given density of the fluid ρ, and
assuming that the external atmospheric pressure has stayed constant?
Also, we're using Imperial units for this problem:
- Pressure is given in units of
psi - Density is given in
lb/ft^3 - Distance is given in
inches - For acceleration due to gravity, use
g = 32.2 ft/s^2
Input Data
First line is Q, the quantity of testcases.
Q lines will then follow, each holding three space-separated decimal values: P_A, P_B, and ρ.
Answer
Should consist of Q space-separated values corresponding to distance that h_B has shifted.
Report downward shifts as negative distances.
Error should be less than 1e-6.
Example
input data:
1
14.700 14.702 62.4
answer:
0.00086