Calculating Moment of Inertia of Long, Thin Rod w/ Varying Mass Density

[factor]

I've been trying to find the moment of inertia of a long thin rod of mass M and length L whose mass density increases as the square of the distance from the axis which is at one end of the rod and perpendicular to the rod. But so far, I'm pretty stumped on how to do it. My first instinct was using the geometry of the object but that didn't seem to work, now I'm trying to find the centroid of the object's mass distribution to see if that fits any better. Any help on this would be appreciated.

 

[Galileo]

The moment of inertia of a mass element of the rod is x^2dm, where dm is the mass and x the distance from the axis. Just add all the contributions up (i.e. integrate) and use dm=rho*dx, where rho is the mass density (mass/unit lenght).

 

[kyle8921]

Hello,

Calculating the moment of inertia for a long, thin rod with varying mass density can be a challenging task. Your approach of using the geometry of the object is a good start, but as you have noticed, it may not provide an accurate result due to the varying mass density.

One way to approach this problem is by using the parallel axis theorem, which states that the moment of inertia of an object can be calculated by adding the moment of inertia of the object about its center of mass and the product of its mass and the square of the distance between its center of mass and the axis of rotation. In this case, the axis of rotation would be at one end of the rod.

To determine the moment of inertia of the rod about its center of mass, you can divide the rod into infinitesimally small segments and use the formula for the moment of inertia of a thin rod (I=1/12 * mL^2) for each segment. Then, you can use the mass density function (ρ= kx^2) to calculate the mass of each segment and the distance between its center of mass and the axis of rotation. Finally, you can use the parallel axis theorem to calculate the moment of inertia of the entire rod.

Another approach would be to use the integral formula for moment of inertia, which takes into account the varying mass density. This method involves integrating the mass density function over the length of the rod and using the result as the mass in the moment of inertia formula. However, this approach may be more complex and time-consuming.

I hope this helps and provides some direction for your calculations. If you need further assistance, please do not hesitate to reach out. Good luck with your research!