Where is the Center of Mass in a Solid Hemisphere?

In summary, the center of mass is the average of the masses factored by their distances from a reference point, which in this case is the center of the hemisphere.
  • #1
Westin
87
0

Homework Statement


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

z=0
0
char3C.png
z
char3C.png
R
char3D.png
2
z=R
char3D.png
2
R
char3D.png
2
char3C.png
z
char3C.png
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]
 

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  • #2
Westin said:

Homework Equations

None
Really? No equation for the center of mass?

Westin said:
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.
 
  • #3
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...
 
  • #4
NTW said:
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.
 
  • #5
DrClaude said:
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.
 
  • #6
NTW said:
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.
 

Related to Where is the Center of Mass in a Solid Hemisphere?

What is the hemisphere center of mass?

The hemisphere center of mass is the point at which the entire mass of a hemisphere is considered to be concentrated. It is the average location of all the individual particles in the hemisphere, taking into account their mass and distribution.

How is the hemisphere center of mass calculated?

The center of mass of a hemisphere can be calculated using the formula: x = 0, y = 0, z = (3R/8), where R is the radius of the hemisphere. This means that the center of mass is located at the center of the flat base of the hemisphere, and at a distance of 3/8 of the radius from the base to the top.

Why is the hemisphere center of mass important?

The center of mass is important because it is the point at which an object is perfectly balanced. In the case of a hemisphere, knowing the center of mass is crucial for understanding its stability and how it will behave when subjected to external forces.

Does the center of mass change with the size of the hemisphere?

Yes, the center of mass changes with the size of the hemisphere. As the radius increases, the center of mass moves further away from the base and closer to the top of the hemisphere. This is because the distribution of mass becomes more spread out as the size increases.

Can the center of mass be outside of the hemisphere?

No, the center of mass will always be located within the hemisphere. This is because the distribution of mass in a hemisphere is symmetrical and there is no mass outside of the hemisphere to affect the center of mass calculation.

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