Finding the Center of Mass of a Hollowed Out Sphere

In summary, the center of mass of the new sphere is located at (1/6) of the way from the center of the stryofoam sphere.
  • #1
Blingo
3
0
Hey guys, this is my first thread here. I'm just looking for a hint or two (no answer please).


A sphere of styrofoam has radius R. A cavity of radius R/2 centered a distance R/2 directly above the center of the sphere is hollowed out and filled with a solid material of density five times the density of styrofoam. Where is the center of mass of the new sphere?


I figured that to do three dimensional center of mass calculations, I just break it down for each dimension and solve, so I was able to get the center of mass for the pre-cavity styrofoam sphere, which is just R.

I think that, if the new sphere's center of mass were to be calculated in the same manner, the mass density would no longer be constant and would therefore be a function. However, that function is eluding me.
 
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  • #2
Blingo said:
Hey guys, this is my first thread here. I'm just looking for a hint or two (no answer please).

Maybe, you could treat this situation as two constant density spheres instead of one sphere and one sphere with a hollow somehow?
 
  • #3
hmm, thanks. I'll try that now.
 
  • #4
Treat it first as three spheres:
the stryofoam sphere complete - density 'rho', the filled cavity density 5'rho' and a 'subtracted' cavity of density -rho.

In other words treat it as two spheres: the complete stryofoam sphere and a filled cavity of 4'rho'.

Garth
 
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  • #5
thanks guys for the help. one final question -- when i treat the problem as two separate spheres, i thought to use the equation


Y = (m1y1 + m2y2)/(m1 + m2)


however, i don't know how to determine the distance between the two spheres. i determined the masses by multiplying the density by volume, but perhaps that ain't the way to do it.
 
  • #6
R/2 - [Edit] draw a diagram it is easy. Remember the mass of the smaller sphere is reduced by the amount of stryofoam you have taken out.

mass is density times volume - hint: density is mass/volume!
 
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  • #7
Blingo said:
Y = (m1y1 + m2y2)/(m1 + m2)
That's fine. (I would measure the distances from the center of the large sphere.)

however, i don't know how to determine the distance between the two spheres.
It's given in the statement of the problem: "A cavity of radius R/2 centered a distance R/2 directly above the center of the sphere..."

i determined the masses by multiplying the density by volume, but perhaps that ain't the way to do it.
That's fine. All you need is the ratio of the masses.
 
  • #8
Thanks!

I think I've got it now.


The volume of the first sphere is:

V1 = (4/3)(pi)R^3

The volume of the second sphere is:

V2 = (4/3)(pi)(R/2)^3 = (1/6)(pi)R^3

Comparing the two spheres gives:

V1 = 8 * V2




The mass of the first sphere is:

M1 = Rho * V1

The mass of the second sphere is:

M2 = (4 * Rho) * V2 = (4 * Rho)(V1/8)

Once again, comparing the two spheres gives:

M2 = (1/2) M1



Then using the equation

Y = [(M1 * Y1) + (M2 * Y2)]/(M1 + M2) = (M2 * Y2)/(M1 + M2)

Y = [(1/2)M1 * (R/2)]/[M1 + (1/2)M1] = [(1/2)M1 * (R/2)]/[(3/2)M1]

Y = (1/6)R

Which is what the answer book says.
 

What is the Center of Mass Problem?

The Center of Mass Problem is a physics problem that involves finding the point at which an object's mass is evenly distributed in all directions. It is also known as the Center of Gravity or Barycenter.

Why is the Center of Mass important?

The Center of Mass is important because it allows us to understand the overall motion and stability of an object. It also helps us calculate the forces acting on an object and predict how it will respond to those forces.

How is the Center of Mass calculated?

The Center of Mass can be calculated by finding the weighted average of the individual masses and their distances from a reference point. This can be done mathematically using integration or graphically using a balance scale or plumb line.

Is the Center of Mass always located within the object?

No, the Center of Mass can be located both inside and outside of an object. It depends on the distribution of mass within the object. For example, a donut-shaped object will have its Center of Mass located in the empty space in the center.

How does the Center of Mass change when an object moves or rotates?

The Center of Mass will always remain at the same point in an object, regardless of its motion or orientation. However, the position of the Center of Mass relative to the reference point may change as the object moves or rotates, affecting its overall stability and balance.

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