Buoyant Force acting on a sphere

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Homework Help Overview

The problem involves a sphere floating in water, specifically focusing on the calculation of the buoyant force acting on it. The sphere has a radius of 10.0 cm and is partially submerged, with its lowest point 5.00 cm below the water's surface.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the formula for buoyant force and the volume of the submerged part of the sphere. There are attempts to calculate the buoyant force using different interpretations of the submerged volume, including confusion over whether to use the full radius or half of it.

Discussion Status

Participants are actively engaging with the problem, checking arithmetic and questioning their assumptions about the submerged volume. Some have suggested sketching the scenario to clarify their understanding, and there is recognition of potential misinterpretations in the initial calculations.

Contextual Notes

There is a mention of differing answers between participants and the textbook, indicating a need for careful consideration of the calculations and assumptions regarding the submerged volume of the sphere.

Carrie
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Homework Statement


A sphere of radius 10.0 cm floats in equilibrium partially submerged in water with its lowest point 5.00 cm below the water's surface.

(a) What is the buoyant force acting on the sphere?

Homework Equations


F = pvg

The Attempt at a Solution


F= 1000 * v * 9.8

V= 4/3*pi*(r)^3 and then I multiplied this by 1/2 because half of it is submerged.

F= 1000 * 1/2(4/3*pi*(.1)^3) * 9.8
F= 202 N, but the answer is 6.42 N, so that is very wrong.Thank you!
 
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Carrie said:

Homework Statement


A sphere of radius 10.0 cm floats in equilibrium partially submerged in water with its lowest point 5.00 cm below the water's surface.

(a) What is the buoyant force acting on the sphere?

Homework Equations


F = pvg

The Attempt at a Solution


F= 1000 * v * 9.8

V= 4/3*pi*(r)^3 and then I multiplied this by 1/2 because half of it is submerged.

F= 1000 * 1/2(4/3*pi*(.1)^3) * 9.8
F= 202 N, but the answer is 6.42 N, so that is very wrong.Thank you!
Check your arithmetic. I don't get your answer or the book's answer, either.
 
SteamKing said:
Check your arithmetic. I don't get your answer or the book's answer, either.
I think I might have messed up putting it in my calculator, because now I'm getting 20.5 N.
 
The sphere's radius is 10 cm. The lowest point is 5 cm below the water's surface. Sketch it.
 
Carrie said:
I think I might have messed up putting it in my calculator, because now I'm getting 20.5 N.
Closer, but still not what I calculated.
 
Okay, I sketched it and now I'm wondering if I use half that radius of 10 cm because it's half submerged. So:

F= 1000 * 1/2(4/3*pi*(.05)^3) * 9.8 = 5.13 N. I'm not sure if I'm going off in the wrong direction now.
 
Carrie said:
Okay, I sketched it and now I'm wondering if I use half that radius of 10 cm because it's half submerged. So:

F= 1000 * 1/2(4/3*pi*(.05)^3) * 9.8 = 5.13 N. I'm not sure if I'm going off in the wrong direction now.
No, the sketch is very helpful here. You want to find the volume of part of a sphere, what is called a spherical cap, like this (the blue region would be submerged):

15e369cd-befe-4b1c-bc5e-269ae7a025a7.png
https://en.wikipedia.org/wiki/Spherical_cap

I think I misread the problem like you did in my initial attempt at calculation, so the book answer could be correct.
 
Oh, that makes sense! Thank you for your help!
 

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