Moment of Inertia through center of mass

[eprparadox]

Homework Statement

A rod is 10 ft long and has a density that goes from 4 to 24 lb/ft.

a) Find the mass.
b) Find the center of mass.
c) Find the moment of inertia through the center of mass.

Homework Equations

I = Ml^2
I = I(cm) + Md^2 (parallel axis theorem)

The Attempt at a Solution

So I got figured out the solution to this problem.

a) M = 140
b) x(cm) = 6.19
c) I(cm) = 6.92M

My question revolves around part c. The way I found it was to use the parallel axis theorem. I found the moment of inertia at one end of the rod. Then I subtracted M*d^2 where d= 6.19 (the distance to the center of mass. This gave me the correct answer.

However, I was trying for some time to just get the moment of inertia about the center of mass directly from the definition of moment of inertia, but I couldn't get the integration limits right.

If I have my axis at the center of mass, I don't know how to setup the integral properly. Or is it that you have to use the parallel axis theorem?

Any help would be greatly appreciated.

 

[SteamKing]

1. You are given units in the OP. Your answers should also reflect the proper units.

2. Your description of the calculation of the moment of inertia is not clear.

3. One way to calculate the moment of inertia is to set up the inertia calculation using one end of the rod as the initial reference point for both the first and second moments of mass. Once these moments are determined about the endpoint, then the PAT can be used to find the moment of inertia about the COM. There should be no confusion about the limits of integration in this case.

 

[mfb]

but I couldn't get the integration limits right.

You can still integrate from 0 to 10, just take the squared distance to the center of mass instead of the position as integrand (multiplied by the density of course).

Or is it that you have to use the parallel axis theorem?

You don't have to.

Your answers have missing units.

 

[eprparadox]

1. In regards to the units, that's my bad. In haste, I wrote the answers without units.

M = 140 lb
x(cm) = 6.19 ft.

2. I found my error when calculating the moment of inertia about the center of mass. Just as mfb stated, the l^2 term will have to be the distance from the center of mass point, which is (6.19 - l)^2. This would be, of course, multiplied by the density and then integrated from 0 to 10 for the correct answer.

Thanks so much!

 

[Nickie]

Dear student,

First of all, great job on finding the solutions to parts a) and b) of the problem. As for part c), you were on the right track with using the parallel axis theorem. The reason you couldn't get the moment of inertia directly from the definition is because the density of the rod is not constant throughout its length. This means that the mass distribution is not uniform and therefore, the moment of inertia cannot be calculated using the standard formula (I = ∫r^2dm).

To calculate the moment of inertia directly from the definition, you would need to break the rod into infinitesimally small sections and integrate the moment of inertia of each section (using the formula I = ∫r^2dm). However, this would be a very tedious and time-consuming process.

Using the parallel axis theorem is a much simpler and more efficient way to calculate the moment of inertia in this case. It allows you to take into account the varying density of the rod and still get the correct answer. So, in conclusion, you were on the right track and using the parallel axis theorem is the correct approach to finding the moment of inertia through the center of mass in this problem. Keep up the good work!