Find Center of Mass of Hemisphere Homework

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In summary, the center of mass for a hemisphere of radius R is found by using the equation Cm=(1/m)\int(z dm) with dm = \rho A dz, where A is the area of the circular disc and ρ is the volume density. The final integral is \frac{1}{M} \int_0^R \ dz\ \pi (R^2 - z^2) \rho\ z, which gives the result \frac{3 R}{8}.
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JJfortherear
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Homework Statement


A hemisphere of radius R, covering +/- R in the x and y directions and 0 to R in the Z direction (only the top half of a sphere centered at the origin)
Find the center of mass.

Homework Equations


Cm=(1/m)[tex]\int(z dm)[/tex]

The Attempt at a Solution


dm is sigma dA, and dA is x dz, where x = [tex]\sqrt{}R^2-z^2[/tex]. The limits are 0 to R, and the answer should be 3/8 R, but when I evaluate the integral I get nothing like that.
 
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  • #2
since you are trying to find for a hemisphere not for a hemispherical shell, your equation for 'dm' is wrong.
You should consider not the surface density, but instead the volume density ρ since it is a solid object
[itex] \rho = \frac{M}{\frac{2}{3} \pi R^3} [/itex]

You then have

[itex] dm = \rho A dz [/itex] where A is the area of the circular disc centered on the z-axis with radius [itex] r = \sqrt{R^2 - z^2} [/itex].

The final integral then becomes

[itex]CM (z-comp) = \frac{1}{M} \int_0^R \ dz\ \pi (R^2 - z^2) \rho\ z [/itex]

which on solving gives the result [tex] \frac{3 R}{8} [/tex]
 

What is the center of mass of a hemisphere?

The center of mass of a hemisphere is the point at which the mass of the hemisphere is evenly distributed in all directions. It is also known as the centroid or center of gravity.

Why is it important to find the center of mass of a hemisphere?

Finding the center of mass of a hemisphere is important in many fields, such as physics, engineering, and architecture. It helps in predicting the behavior of the hemisphere when subjected to external forces, and in designing structures that are balanced and stable.

How do you calculate the center of mass of a hemisphere?

The center of mass of a hemisphere can be calculated by using the formula x = 0, y = 0, z = (3/8)r, where r is the radius of the hemisphere. This formula assumes that the density of the hemisphere is constant and the center of mass lies on the z-axis.

Can the center of mass of a hemisphere be outside of the hemisphere?

No, the center of mass of a hemisphere will always lie within the hemisphere. This is because the mass of the hemisphere is evenly distributed and there is no external force acting on it.

What are some real-life applications of finding the center of mass of a hemisphere?

Some examples of real-life applications include balancing objects such as a bowling ball or a satellite, designing structures such as arches and domes, and predicting the motion of a spinning top or a spinning wheel.

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