# Minimum force required to form sphere

Chestermiller
Mentor
Hello,

Could you please explain as to why the force exerted by external atmospheric pressure acting on the outer curved surface of the hemisphere would be PatmπR2 ?
The external atmospheric pressure acts perpendicular to the outer curved surface at each location on the surface. See what you get if you integrate this (vectorially) over the entire curved surface area of the hemisphere.

Chet

• Tanya Sharma
TSny
Homework Helper
Gold Member
Could you please explain as to why the force exerted by external atmospheric pressure acting on the outer curved surface of the hemisphere would be PatmπR2 ?
Chet's method will nail it.

A conceptual argument can be made that might or might not satisfy you. Imagine the atmosphere as a static fluid. Consider a hemispherical volume of the atmosphere with the flat surface of the volume oriented vertically. Consider the forces acting on the flat and curved surfaces of the volume due to the surrounding atmosphere. Argue that these forces must add to zero.

• Chestermiller
The external atmospheric pressure acts perpendicular to the outer curved surface at each location on the surface. See what you get if you integrate this (vectorially) over the entire curved surface area of the hemisphere.

Chet

Do you mind showing how you would arrive at the result using above approach .

A conceptual argument can be made that might or might not satisfy you. Imagine the atmosphere as a static fluid. Consider a hemispherical volume of the atmosphere with the flat surface of the volume oriented vertically. Consider the forces acting on the flat and curved surfaces of the volume due to the surrounding atmosphere. Argue that these forces must add to zero.

Yes ,that makes sense if we have same fluid on the flat and curved surfaces . But in the OP we have different fluids .

Can you think of an argument considering only the curved surface just like you did in post#8 ?

TSny
Homework Helper
Gold Member
Yes ,that makes sense if we have same fluid on the flat and curved surfaces . But in the OP we have different fluids .

The external atmospheric force acting on one of the hemispherical curved surfaces of the sphere filled with water is independent of what fluid is inside the sphere.

The left figure below shows a region of the atmosphere that has the same shape as a hemisphere of the sphere filled with water. The arrows indicate the external atmospheric force acting on the flat and curved surfaces of this portion of the atmosphere. The figure on the right is the sphere filled with water and the arrow indicates the external atmospheric force acting on the curved portion of one of the hemispheres of the sphere. All three forces F are equal. The force on the far left is easy to calculate.

#### Attachments

• Hemisphere in atmosphere.png
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• Tanya Sharma and Chestermiller
Chestermiller
Mentor
Do you mind showing how you would arrive at the result using above approach .
Well. Let me get you started. The force per unit area exerted by the atmosphere on the curved surface of the sphere is ##-p_{atm}\vec{i}_r##, where ##\vec{i}_r## is the unit vector in the radial direction in spherical coordinates. Do you know how to determine the x component of this unit vector in terms of the spherical coordinate angles φ and θ? Do you know how to determine the differential element of area on the surface of the sphere in terms of R, dφ, and dθ?

Chet

• Tanya Sharma
The external atmospheric force acting on one of the hemispherical curved surfaces of the sphere filled with water is independent of what fluid is inside the sphere.

The left figure below shows a region of the atmosphere that has the same shape as a hemisphere of the sphere filled with water. The arrows indicate the external atmospheric force acting on the flat and curved surfaces of this portion of the atmosphere. The figure on the right is the sphere filled with water and the arrow indicates the external atmospheric force acting on the curved portion of one of the hemispheres of the sphere. All three forces F are equal. The force on the far left is easy to calculate.

Thanks .

Here is what I had been thinking .

Let the flat surface be in the z plane . The force on a very small surface element will be ##pd\vec{A}## . This ##d\vec{A}## could be decomposed into three area elements along the three planes . Now we could find another element on the surface such that the two elements ##d\vec{A_x}## and ##d\vec{A_y}## get cancelled . So that we are only left with ##d\vec{A_z}## . Now ##\int d\vec{A_z} = \pi R^2## .

I am not sure if I could express myself properly . Does it make any sense ?

TSny
Homework Helper
Gold Member
Let the flat surface be in the z plane . The force on a very small surface element will be ##pd\vec{A}## . This ##d\vec{A}## could be decomposed into three area elements along the three planes . Now we could find another element on the surface such that the two elements ##d\vec{A_x}## and ##d\vec{A_y}## get cancelled . So that we are only left with ##d\vec{A_z}## . Now ##\int d\vec{A_z} = \pi R^2## .

I am not sure if I could express myself properly . Does it make any sense ?

Yes. That sounds very good.

• Tanya Sharma
Yes. That sounds very good.

Hard to believe that you liked my reasoning .

Well. Let me get you started. The force per unit area exerted by the atmosphere on the curved surface of the sphere is ##-p_{atm}\vec{i}_r##, where ##\vec{i}_r## is the unit vector in the radial direction in spherical coordinates. Do you know how to determine the x component of this unit vector in terms of the spherical coordinate angles φ and θ? Do you know how to determine the differential element of area on the surface of the sphere in terms of R, dφ, and dθ?

Chet

Thank you Chet .

Sorry , but as of now I do not possess required mathematical skills .

TSny
Homework Helper
Gold Member
Hard to believe that you liked my reasoning .
I know. Probably some aftereffect of the eclipse of the moon last night. • Tanya Sharma
I know. Probably some aftereffect of the eclipse of the moon last night.  