Buoyant Force acting on a sphere

[Carrie]

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!

 

[SteamKing]

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.

 

[Carrie]

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.

 

[gneill]

The sphere's radius is 10 cm. The lowest point is 5 cm below the water's surface. Sketch it.

 

[SteamKing]

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.

 

[Carrie]

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.

 

[SteamKing]

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):

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.

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[Carrie]

Oh, that makes sense! Thank you for your help!