Finding COM, and inertia my answers don't make sense?

[amyparker30]

Finding COM, and inertia please help my answers don't make sense??

Homework Statement

Find the center of mass of the object shown in the figure below?

LINK TO FIGURE:
http://s1168.photobucket.com/albums/r497/amy_parker1/?action=view&current=Untitled.jpg

1. Find the center of mass of the object shown in the figure below.

2. Calculate the rotational inertia of the object about the x-axis.

3. From this value, deduce the rotational inertia of the object about an axis parallel to the x-axis, and going through the center of mass.

m1=2.5kg , m2=5kg, m3=2.5kg , m4=5kg

Homework Equations

COM

Inertia?

The Attempt at a Solution

my answers don't make sense

for COM I got 2

 

[NascentOxygen]

Hi amyparker30! Where are your answers? Where is your working?

 

[amyparker30]

ok, my answers are totaly wrong but for COM 1 got (2(2.5+5+2.5+5))/15 = 2

ICOM=Rotational Inertia = 2^2(2.5+5+5+2.5) = 60

i bet these are 100% wrong what should I do, my teacher wouldn't help me and the people from class just skipped this question :(

 

[joseamck]

Ok first you need to understand the COM formula and see how it works. Basically you need to specify a coordinate to tell where the COM is. That means you need to find the center of mass in the x direction, y direction, and z direction of the system or object.
So we start by separating everything in components( x, y and z). Let's do the x direction first.

We let $d_{nx}$ represent the distance in the x direction for each particle (n=1,2,3,4)
Then
$X_{com} = \frac{m_2 d_{2x}+m_3 d_{3x} + m_1 d_{1x}+m_4 d_{4x}}{M}$
where M is the total mass. Note that
$d_{2x}=d_{1x}=0$
and
$d_{3x}=d_{4x}=2 m$

Similarly for y:
$Y_{com} = \frac{m_2 d_{2y}+m_3 d_{3y} + m_1 d_{1y}+m_4 d_{4y}}{M}$
Note that
$d_{2y}=d_{3y}=0$
$d_{1y}=d_{4y} = 2 m$

Similarly for z:
$Z_{com} = \frac{m_2 d_{2z}+m_3 d_{3z} + m_1 d_{1z}+m_4 d_{4z}}{M}$
But Note that
$d_{1z} = d_{2z} = d_{3z} = d_{4z} = 0$
then
$Z_{com} = 0$

Then at the end you get the COM to be at $(X_{com},Y_{com},Z_{com})$

Now try to plug in the numbers and see what you get. (Easy! =))

Do the same thing for the next part (moment of inertia), first understand the formula/equation and just follow it slowly.

 

[NascentOxygen]

I think the first part of this question would be a good one for a multiple choice test.

By symmetry, the CoM of the two equal masses m1 and m2 is mid-way between them. Likewise, the CoM of the equal mass pair m3 and m4 is mid-way between m3 and m4. And you wouldn't believe our luck but those two CoM locations coincide!

So that surely must be the CoM for the 4 bodies.