Change the chapter
Question
(a) How high will water rise in a glass capillary tube with a 0.500-mm radius? (b) How much gravitational potential energy does the water gain? (c) Discuss possible sources of this energy.
1. $2.97 \textrm{ cm}$
2. $1.70 \times 10^{-6} \textrm{ J}$
3. Work is done by the surface tension. This work causes the change in gravitational potential energy.
Solution Video

# OpenStax College Physics for AP® Courses Solution, Chapter 11, Problem 79 (Problems & Exercises) (2:42)

Rating

No votes have been submitted yet.

Quiz Mode

Why is this button here? Quiz Mode is a chance to try solving the problem first on your own before viewing the solution. One of the following will probably happen:

1. You get the answer. Congratulations! It feels good! There might still be more to learn, and you might enjoy comparing your problem solving approach to the best practices demonstrated in the solution video.
2. You don't get the answer. This is OK! In fact it's awesome, despite the difficult feelings you might have about it. When you don't get the answer, your mind is ready for learning. Think about how much you really want the solution! Your mind will gobble it up when it sees it. Attempting the problem is like trying to assemble the pieces of a puzzle. If you don't get the answer, the gaps in the puzzle are questions that are ready and searching to be filled. This is an active process, where your mind is turned on - learning will happen!
If you wish to show the answer immediately without having to click "Reveal Answer", you may . Quiz Mode is disabled by default, but you can check the Enable Quiz Mode checkbox when editing your profile to re-enable it any time you want. College Physics Answers cares a lot about academic integrity. Quiz Mode is encouragement to use the solutions in a way that is most beneficial for your learning.

## Calculator Screenshots

Video Transcript
This is College Physics Answers with Shaun Dychko. The height that the water will rise up this tube due to the capillary action is two times the water's surface tension times cosine of the contact angle between the water and the material the tube is made out of which is glass, divided by the density of water times g times the radius the tube. So that's two times 0.0728 newtons per meter surface tension for water, times cosine of zero, divided by the density of water, one times ten to the three kilograms per cubic meter, times 9.8 times 0.5 times ten to the minus three meters, radius of the tube. So it will rise to a height of 2.97 centimeters. The next question is what will be the gravitational potential energy gained by the water through it being raised to this height? Well, we're going to multiply the weight of the water m g by the average height that it rises because the water will rise to a maximum h but it has this meniscus here. It has this u-shape to it so not all of the water in this column gets raised to this maximum height. So we look for some average that will have an equal amount of water above the average that is equal to the amount of air below the average. This average looks like it's about half the height. So we'll say eight over two is our average height. So then we want to find out what the mass of water that's been raised is and so we have density is mass divided by volume and then multiply both sides by volume and you solve for m. It's the density times the volume. The volume will be the cross-sectional area of this water column multiplied by its average height h over two. So then we substitute all of this in for m in our potential energy formula. So we have rho pi r squared h over two times g times h over two. So that's rho pi r squared h squared over four. So we have pi over four here times the density of water times g times the radius of the tube squared, times 0.02968 meters the height of the water column, and then square that. We get 1.7 times ten to the minus six joules. This potential energy comes from a conservative force as all potential energies do. Potential energy comes from the work done by a conservative force and in this case the conservative force is the surface tension. So that's where the gravitational potential comes from, it's from work done by the surface tension.