Change the chapter
Question
Referring to Figure 11.20, prove that the buoyant force on the cylinder is equal to the weight of the fluid displaced (Archimedes’ principle). You may assume that the buoyant force is $F_2 - F_1$ and that the ends of the cylinder have equal areas $A$. Note that the volume of the cylinder (and that of the fluid it displaces) equals $(h_2 - h_1)A$.
Question Image
<b>Figure 11.20</b> An object submerged in a fluid experiences a buoyant force.
Figure 11.20 An object submerged in a fluid experiences a buoyant force.
Question by OpenStax is licensed under CC BY 4.0.
Please see the solution video.
Solution Video

OpenStax College Physics for AP® Courses Solution, Chapter 11, Problem 52 (Problems & Exercises) (2:27)

Sign up to view this solution video!

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.

Video Transcript
This is College Physics Answers with Shaun Dychko. We are going to prove that the buoyant force is the weight of fluid displaced by the object and this is Archimedes principle. So we are told to assume that the top and bottom face of the cylinder are of equal areas A and there is force F 2 upwards here and then force F 1 downwards at the top surface and this difference between these two forces is the buoyant force. So this F 2 is greater since the pressure deeper is higher and whereas here, this surface is at a shallower depth and so its pressure is less there and so there's less force. So the buoyant force is that difference and F 2 is gonna be the pressure at this bottom depth multiplied by the area and then the force on the top face will be the pressure at that depth, P 1, times that same area A and we can factor out the area and the pressure is the density of the fluid times gravitational field strength times the height so P 2 is ρgh 2 and P 1 is ρgh 1 and we can substitute for each of those now on this line here and this ρ and the g can be factored out. So we have area times density of the fluid times g times the difference in heights. Now the volume of the object is equal to the volume of the fluid displaced because this object is in the water now completely, or whatever fluid it is, it's completely in the fluid and it's going to displace volume of fluid equal to the volume of the object. And the volume of this object is its area multiplied by its height— that's the volume of a cylinder; it's the area of the circular face multiplied by the height of the cylinder. So we can substitute that in place of A times h 2 minus h 1 and so the buoyant force then is the volume of the fluid times the density times g and the volume times the density is the mass and it's being multiplied by g here and so that is the weight of the fluid displaced and therefore, we have shown that the buoyant force is the weight of the fluid displaced and that is Archimedes principle.