Where is the Center of Mass in a Solid Hemisphere?

[Westin]

Homework Statement

Consider a solid hemisphere of uniform density with radius R. Where is the center of mass?

z=0
0

z

R

2
z=R

2
R

2

z

r
z=R

Image is provided.

Homework Equations

None

The Attempt at a Solution

Answer A and E do not seem logical. I thought it was answer C from my eyes. Center of mass is the average of the masses factored by their distances from a reference point. I didn't think the answer could range like B and D do.[/B]

 

[DrClaude]

Homework Equations

None

Really? No equation for the center of mass?

Answer A and E do not seem logical. I thought it was answer C from my eyes. Center of mass is the average of the masses factored by their distances from a reference point. I didn't think the answer could range like B and D do.

Do you understand that the range is there to keep you from having to calculate the exact value? It doesn't mean that the center of mass can be anywhere within that range.

 

[NTW]

If the center of mass were precisely at R/2, there will be more mass below than above that point. Hence, it must be somewhere between R/2 and...

 

[DrClaude]

If the center of mass were precisely at R/2, there will be more mass below than above that point.

I don't understand what you mean.

 

[NTW]

I don't understand what you mean.

To make a mental experiment: If I imagine a given point on the Z axis, precisely at R/2, and also imagine the hemisphere as formed by a very large, but finite number of particles, the number of particles with z-coordinates lower than R/2 will be larger than the number of particles with z-coordinates higher than R/2. Hence, in order to reach a 50% partition in the values of the z-coordinates, that point must be placed somewhere between 0 and R/2.

 

[DrClaude]

If I imagine a given point on the Z axis, precisely at R/2

Ok, but that's not the same as saying "if the center of mass were precisely at R/2."

Also, please to not give direct answers in the homework forums. The poster has to do the work.