Homework Help: Center of Mass of open top cylinder

1. Jul 25, 2015

naianator

1. The problem statement, all variables and given/known data

The attached diagram shows a uniform density hollow cylindrical shell with a solid bottom and an open top. It has radius R and height h.

Find the height for the center of mass of this cylinder, taking the origin of the coordinate system at the center of the bottom. Use "pi" for π.

2. Relevant equations
x_cm=m_1x_1+m_2x_2+m_3x_3/m_1+m_2+m_3

3. The attempt at a solution
I'm not even really sure how to start but I tried this

If the top wasn't missing then the cylinder would have a center of mass at h/2 and the missing top has a center of mass at h so

CM = (h/2*(2*pi*R*h+2*pi*R^2)-h*pi*R^2)/2*pi*R*h+2*pi*R^2-pi*R^2

= pi*R*h^2/2*pi*R*h-pi*R^2

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2. Jul 25, 2015

Borg

If you had just the top piece located at height h, could you determine the center of its mass?

3. Jul 25, 2015

naianator

Wouldn't it just be h?

4. Jul 25, 2015

Borg

And can you determine the mass of it?

5. Jul 25, 2015

naianator

Wouldn't the masses cancel though? h = h*m/m?

6. Jul 25, 2015

haruspex

You really ought to parenthesise expressions correctly. You mean
(h/2*(2*pi*R*h+2*pi*R^2)-h*pi*R^2)/(2*pi*R*h+2*pi*R^2-pi*R^2)
or in LaTex
$\frac{\frac h2(2\pi Rh+2\pi R^2)-h\pi R^2}{2\pi Rh+2\pi R^2-\pi R^2}$
But you made a mistake in simplifying to
$\frac{\pi Rh^2}{2\pi R h-\pi R^2}$
(Note that that would make it > h/2.)

7. Jul 25, 2015

naianator

I'm having trouble finding where the mistake is. Is the first expression correct?

8. Jul 25, 2015

haruspex

Yes, it's just the last line that's wrong. Check the signs.

9. Jul 25, 2015

naianator

Ahhh yes! Thank you

10. Jul 25, 2015

SammyS

Staff Emeritus
Rather than that:

If both the top and bottom were missing, then the CM would be at h/2.

To this add in the bottom, which has CM at 0 .

It makes the algebra a bit easier.