# Homework Help: Moment of Inertia of two small bars

1. Apr 21, 2008

### jaredmt

1. The problem statement, all variables and given/known data
A uniform bar has two small balls glued to its ends. The bar is 2m long and has mass 4kg, while the balls each have mass .5kg and can be treated as point masses. Find the moment of inertia of this combination about each of the following axes: (a) an axis perpendicular to the bar through its center; (b) an axis perpendicular to the bar though one of the balls; (c) an axis parallel to the bar through both balls; (d) an axis parallel to the bar and .5m from it.

3. The attempt at a solution

i tried the formula: (1/12)(ML^2) but that got the wrong answer and i assume because the balls at the end changed it. im not sure what to do... any help would be very much appreciated

2. Apr 21, 2008

### Lord Crc

Yes, the balls change the total moment of inertia. If you take a look at how the moment of inertia is defined for a rigid body of point masses then you should find the answers to your questions. If you don't find it in your book, I can recommend wikipedia :)

3. Apr 21, 2008

### jaredmt

ok i beleive i found the answers but im not entirely sure if i did it all correctly so please correct me if my proceedure was wrong:

a) I = (1/12)(4kg)(2m)^2 + 2(.5kg)(1m)^2 = 1.33 + 1 = 2.33 kgm^2
the first part i got the formula as i described above. then for the last part: 2 is because there is a total of 2 balls (1 on each side) and .5kg is the mass of each then 1m is the distance the balls are from the center

b) I = (1/3)(4kg)(2m)^2 + .5kg(2m)^2 = 5.33 + 2 = 7.33 kgm^2
the first part i got from this formula: (1/3)(mass)(distance)^2
the second part(for the ball) i just have as (mass)(distance)^2

also, one more question: is there a way that i could throw the center of mass into a formula to find the moment of inertia? or do i have to break it up into 2 parts like this? because i found that the center of the mass, going from the center-point to the end-point, is .6m from the center. but i plugged it into formulas and it didnt come out right

Last edited: Apr 21, 2008
4. Apr 21, 2008

### CrazyIvan

The Moment of Inertia for N point masses is defined as
$$I = \sum^{N}_{i=1} m_{i} r_{i}^{2}$$

The Center of Mass for N point masses is
$$\vec{R} = \frac{\sum m_{i} \vec{r}_{i}}{\sum m_{i}}$$

Considering that $$\vec{R}$$ is a vector and $$I$$ is a scalar, I can't think of any easy way to define one in terms of the other.

And considering that you are only dealing with 3 masses, it's probably easiest to just chug through the old-fashioned way.

5. Apr 21, 2008

### jaredmt

ok, just makin sure im not taking any long steps when there is a much shorter step. but i dont think finding the cm and plugging in a formula would make it any shorter anyways come to think of it.

but did i find the answers correctly? or did i just somehow luck out with the right answer using the wrong formulas? the 1st parts of each formula seemed right, i just had to plug the numbers in. but the second part, i not 100% sure. i dont see how else i could do it so im 90% sure i have it all right

6. Apr 21, 2008

### CrazyIvan

Both calculations look correct.

Just keep plugging along with that Parallel Axis Theorem