Moment of inertia bar variable density

[chrisandmiss]

Homework Statement

I think I solved this correctly, but I am not sure.

Find the moment of inertia of a bar with density d=kx^2, where x is the distance from the center of the bar(find the moment of inertia at the center)

THe length of the bar is L

Homework Equations

d=kx^2

Moment of inertia=sum of mx^2

The Attempt at a Solution

FInd K. Well, the integral of density over a bar is the mass, so from the center, integrate kx^2 to half of the distance L, gives kx^3/(8*3)= M/2 k= 12M/(L^3)

To find the moment of inertia, sum m1x1^2+m2x2^2...mnxn^2= integral x^2dm

density =kx^2, so dm=kx^2dx

Integrate 2k *x^4(over L/2)which equals 2k* L^5/(32*5)

sub in k

Moment of inertia of the bar equals 3/20 L^2

Is this right? It seems low

 

[anathema]

Thank you for sharing your attempt at solving this problem. It appears that you have correctly calculated the value of k and have correctly set up the integral for finding the moment of inertia. However, there seems to be a mistake in your integration for finding the moment of inertia. The correct integral should be 2kx^4dx, which would give a moment of inertia of 2/5 L^2. This is a fairly low value, but it is possible depending on the density distribution of the bar. I would suggest double checking your integration and if you are still unsure, please feel free to share your work and I can help you identify any mistakes. Keep up the good work!