The cross-sectional area of the U-tube shown below is uniform and is equal to one square centimeter (1.00cm^2). One end is open to the atmosphere. Atmosphere pressure may be taken as P1=1.01 * 10^5 Pa. The other end of the U-tube is connected to a pipe in which a constant gas pressure of P2=1.25 atmospheres is maintained. The U-tube contains a liquid which has a density of 1.0*10^4kg/m^3.
QUESTION: What the second liquid still in the arm that is open to the atmosphere, the pressure p2 is now reduced until it is equal to atmospheric pressure. What now is the difference (if any) between the liquid levels in the two arms of the U-tube?
To determine the difference in liquid levels in the two arms of the U-tube when the gas pressure is equalized to atmospheric pressure, we need to apply the principles of fluid mechanics.
Let's break down the problem step by step:
Step 1: Calculate the pressure difference between the two arms of the U-tube.
Since the U-tube is open to the atmosphere on one end and connected to a pipe with a constant gas pressure on the other end, there is a pressure difference acting on the liquid column in the U-tube.
The pressure difference (∆P) can be calculated using the formula:
∆P = P2 - P1
Where P1 is the atmospheric pressure (1.01 x 10^5 Pa) and P2 is the pressure inside the connected pipe (1.25 x atmospheric pressure).
Given that the pressure in the connected pipe is 1.25 atmospheres, the pressure difference can be calculated as follows:
∆P = (1.25 - 1) x atmospheric pressure
∆P = 0.25 x atmospheric pressure
Step 2: Convert the pressure difference into units of Pa.
Since the pressure difference is given in terms of atmospheric pressure, we need to convert it to Pascal (Pa) for further calculation.
To convert atmospheric pressure to Pa, we use the conversion:
1 atmosphere = 1.01 x 10^5 Pa
Therefore, the pressure difference in Pa can be calculated as:
∆P = 0.25 x 1.01 x 10^5 Pa
∆P = 2.525 x 10^4 Pa
Step 3: Determine the height difference in the liquid columns.
The height difference in the liquid columns can be calculated using the equation of hydrostatic pressure:
∆P = ρgh
Where ρ is the density of the liquid (given as 1.0 x 10^4 kg/m^3), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height difference in meters.
Rearranging the equation to solve for h:
h = ∆P / (ρg)
Substituting the given values:
h = (2.525 x 10^4) Pa / (1.0 x 10^4 kg/m^3 x 9.8 m/s^2)
h ≈ 2.57 meters
Therefore, when the gas pressure is reduced to atmospheric pressure, there will be a height difference of approximately 2.57 meters between the liquid levels in the two arms of the U-tube.