A cylindrical barrel is completely full of water and sealed at the top except fo

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 0.800 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 360 N up. Also, use g = 9.8 m/s^2. The height of water in the tube is cm.

Explanation / Answer

The lid has a diameter of 80cm, so its radius = 0.4m.
Lid's area (ignoring the tube) = r² = x 0.4² = 0.503m²
(The area of the 0.9cm diameter tube is negligible compared to the lid's area).

P = F/A. So a force difference of 270N on the lid means a pressure difference of 270/0.503
=536.77Pa

This pressure difference is due to the extra pressure produced by the column of water in the narrow tube:
height, h, of water in the narrow tube
density of water, = 1000kg/m³
g = 9.8m/s²

P = gh
536.77 = 1000 x 9.8 x h
h = 0.0547m

Height of water in tube is 5.47 cm

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