Chapter 20, Problem 68A

Fill a bottle with water and hold it partially under water so that its mouth is beneath the surface. What happens to the water in the bottle? Explain. How tall would the bottle have to be before water ran out? (Hint: You can’t do this indoors unless you have a ceiling 10.3 m high!)

To determine

To explain: The condition of water in the bottle.

To find: The height of bottle before the water ran out.

Answer to Problem 68A

The bottle with water inside will look like a barometer. The bottle has to be $10.3\text{ m}$ high before water ran out.

Explanation of Solution

Introduction:

The function of barometer is to measure the atmospheric pressure.

When bottle is submerged inside the water with its mouth partially underwater, then water will exert force inside the bottle on the remaining air. It will keep exerting the pressure at a height until it balances with the air pressure inside the bottle. Now, when bottle is pushed deep down then atmospheric pressure decreases and same happens with the air pressure inside the bottle. Water will keep pushing the air inside the bottle until it balances with the air pressure inside the bottle to maintain equilibrium.

Generally, Mercury is used to measure the atmospheric pressure inside the tube. But, if Mercury is replaced with the water then height of the bottle must be $13.6$ times taller than the height of the Mercury column to measure atmospheric pressure which is $13.6\times 0.76\text{ m=}10.3\text{ m}$ . Since $10.3\text{ m}$ column of water weighs equal with the air inside.

Conclusion:

Thus, water will keep pushing the air inside the bottle until it balances each other. This helps in measuring atmospheric pressure as well when height of the bottle is exactly $10.3\text{ m}$ high.