# Finding the Centre of Mass of a Hemisphere

• akrill
In summary: Your method is incorrect. Two equal volumes doesn't work because the center of mass of each piece is a different distance from the plane ##z=h##.
akrill
Homework Statement
Given a hemisphere of radius r and uniform density, find the centre of mass of the hemisphere
Relevant Equations
None given.
• Place hemisphere in xyz coordinates so that the centre of the corresponding sphere is at the origin.
• Then notice that the centre of mass must be at some point on the z axis ( because the 4 sphere segments when cutting along the the xz and xy planes are of equal volume)
• y2 + x2 = r2
• We want two volumes, V1 and V2 which are equal, only cutting parallel to the flat side of the hemisphere at some distance h.
• Recall volume of revolution formula, V = π∫y2dx
• V1 = ∫0h r2 - x2 dx
• Similarly, V2 = ∫hr r2 - x2 dx
• Then, by equating the two integrals and doing some rearrangement, I got to: h3-3r2h +r3 = 0
• Also, 0 < h < r obviously.
Not really sure how to solve for h here, I dont think I made a mistake while rearranging. Is my method valid and/or is there a simpler way to do this?

How do you calculate a moment. And how is moment related to the center of gravity?
If you simplify by taking r=1, then you will be calculating h/r.

Your cubic equation looks correct. How to solve a cubic?

akrill said:
Is my method valid and/or is there a simpler way to do this?
Your method is incorrect. Two equal volumes doesn't work because the center of mass of each piece is a different distance from the plane ##z=h##.

PeroK
PeroK said:
Your cubic equation looks correct. How to solve a cubic?
To be honest I don't think it is right. I've read some more based on what @.Scott said, and I think the correct strategy is to integrate the moments of the "slices" of the hemisphere (thickness dz) and then divide by the mass.

PeroK and .Scott
vela said:
Your method is incorrect. Two equal volumes doesn't work because the center of mass of each piece is a different distance from the plane ##z=h##.
Alright, thanks for clarifying :)

akrill said:
To be honest I don't think it is right. I've read some more based on what @.Scott said, and I think the correct strategy is to integrate the moments of the "slices" of the hemisphere (thickness dz) and then divide by the mass.
Yes, my mistake. I just checked your integration.

In my defence I was cooking and doing physics at the same time.

PhDeezNutz and berkeman
PeroK said:
Your cubic equation looks correct. How to solve a cubic?
BTW, the solution to the cubic equation
$$\alpha^3 -3\alpha +1 = 0$$Where ##0<\alpha <1## is:$$\alpha =2\cos(\frac 4 9 \pi) \approx 0.3473$$

PeroK said:
BTW, the solution to the cubic equation
$$\alpha^3 -3\alpha +1 = 0$$Where ##0<\alpha <1## is:$$\alpha =2\cos(\frac 4 9 \pi) \approx 0.3473$$
Just to be clear, that's the solution to the cubic, but not the center of gravity.

.Scott said:
Just to be clear, that's the solution to the cubic, but not the center of gravity.
If you wanted to divide the hemisphere horizontally into two parts of equal mass, that's the ratio of ##\frac h r## that you would use.

.Scott

## What is the centre of mass of a hemisphere?

The centre of mass of a hemisphere is the point at which the mass of the hemisphere can be considered to be concentrated. For a solid hemisphere, it lies along the vertical axis of symmetry, at a distance of 3R/8 from the flat circular base, where R is the radius of the hemisphere.

## How do you derive the centre of mass of a solid hemisphere?

To derive the centre of mass of a solid hemisphere, you can use calculus and the concept of integration. By considering thin disks of thickness dz along the z-axis and integrating over the volume of the hemisphere, you can calculate the z-coordinate of the centre of mass to be 3R/8 from the flat base.

## Is the centre of mass of a hollow hemisphere different from that of a solid hemisphere?

Yes, the centre of mass of a hollow hemisphere (a hemispherical shell) is different from that of a solid hemisphere. For a hollow hemisphere, the centre of mass is located at a distance of R/2 from the flat circular base along the vertical axis of symmetry.

## Can the centre of mass of a hemisphere be outside the material of the hemisphere?

No, for a solid or hollow hemisphere, the centre of mass is always located within the material of the hemisphere. It lies along the central vertical axis of symmetry, either at 3R/8 or R/2 from the flat base, respectively.

## How does the distribution of mass affect the centre of mass of a hemisphere?

The distribution of mass affects the centre of mass significantly. For a uniform solid hemisphere, the centre of mass is at 3R/8 from the flat base. However, if the hemisphere has a non-uniform density distribution, the centre of mass would shift accordingly, and its exact position would need to be calculated based on the specific density function.

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