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

In summary, the conversation discusses the search for the moment of inertia of a long thin rod with a varying mass density. The approach of using the geometry of the object was unsuccessful, leading to the current attempt of finding the centroid of the mass distribution. The suggestion is to integrate the contributions of each mass element, with the formula x^2dm and using the mass density (rho) to find the mass of each element.
  • #1
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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.
 
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  • #2
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).
 
  • #3


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!
 

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

What is the formula for calculating moment of inertia of a long, thin rod with varying mass density?

The formula for calculating moment of inertia for a long, thin rod with varying mass density is I = ∫r²dm, where I is the moment of inertia, r is the distance from the axis of rotation, and dm is the differential mass.

How do you find the mass density of a long, thin rod?

The mass density of a long, thin rod can be found by dividing the mass of the rod by its length. This will give you the mass per unit length, which is the mass density.

What is the difference between moment of inertia and mass moment of inertia?

Moment of inertia and mass moment of inertia are two different ways of measuring an object's resistance to rotational motion. Moment of inertia is a measure of an object's resistance to angular acceleration, while mass moment of inertia is a measure of an object's resistance to rotational acceleration, taking into account the distribution of mass within the object.

Can the moment of inertia of a long, thin rod with varying mass density be negative?

No, the moment of inertia of a long, thin rod with varying mass density cannot be negative. Moment of inertia is always a positive quantity, as it represents an object's resistance to rotation.

How does changing the mass density along a long, thin rod affect its moment of inertia?

Changing the mass density along a long, thin rod can significantly affect its moment of inertia. The further the mass is from the axis of rotation, the larger the object's moment of inertia will be. This means that if the mass density is higher at one end of the rod, the moment of inertia will also be higher at that end.

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