Question

A cylindrical barrel is completely full of water and sealed at the top except for a narrow tube extending vertically through the lid. The barrel has a diameter of 80.0 cm, while the tube has a diameter of 1.10 cm. You can cause the lid to pop off by pouring a relatively small amount of water into the tube. To what height do you need to add water to the tube to get the lid to pop off the barrel? The lid pops off when the vector sum of the force of the atmosphere pushing down on the top of the lid and the force of the water pushing up on the bottom of the lid is 390 N up. Also, use g = 9.8 m/s2.
What is the height of water in the tube in cm?

Answer 1

To solve this problem it is necessary to apply the concepts related to pressure as a unit that measures the force applied in a specific area as well as pressure as a measurement of the density of the liquid to which it is subjected, its depth and the respective gravity.

The two definitions of pressure can be enclosed under the following equations

$P = \frac{F}{A}$

Where

F= Force

A = Area

$P = \rho gh$

Where,

$\rho =$ Density

g = Gravity

h = Height

Our values are given as,

$d = 0.8m \rightarrow r = 0.4m$

$A = \pi r^2 = \pi * 0.4^2 = 0.503m^2$

If we make a comparison between the lid and the tube, the diameter of the tube becomes negligible.

Matching the two previous expressions we have to

$\frac{F}{A} = \rho g h$

Re-arrange to find h

$h = \frac{F}{A\rho g}$

$h = \frac{390}{(0.503)(1000)(9.8)}$

$h = 0.079m$

$h = 7.9cm$

Therefore the height of water in the tube is 7.9cm