# Moment of inertia of a rod with varying density

• bigevil
In summary, the conversation discusses finding the mass, x-coordinate of the centroid, and moments of inertia of a thin rod with a varying density. The mass is calculated to be 140 lbs, the x-coordinate of the centroid is 130/21, and the moment of inertia about an axis passing through the center of mass is found to be 6.92m. A mistake in the integration limits is identified and corrected.
bigevil

## Homework Statement

A thin rod is 10 ft long and has a density which varies uniformly between 4 and 24 lb/ft. Find:
a) the mass
b) the x-coordinate of centroid
c) Moment of inertia about an axis perpendicular to the rod
d) Moment of inertia about an axis perpendicular to the rod passing thru the heavy end.

## The Attempt at a Solution

a) isn't difficult. I have got $$dm = (2x+4) dx$$, then integrate between 0 and 10 to get 140 lbs.

b) From $$\int \bar{x} dm = \int x dm$$, integrate and use earlier results to get 130/21.

c) I'm stuck at this one. I assume this means that the axis of MI passes through the centre of mass, so I set up the integration limits from 80/21 to -130/21 (rod is 10 foot). Then, taking $$I = \int x^2 dm = \int_{-130/21}^{80/21} 2x^3 + 4x^2 dx = 1.7(140) = 1.7m$$ (where m=140 lbs). But the answer (this question is from Mary L Boas' mathematical methods book) is 6.92m.

Am I missing something here? I am only calculating for the x-coordinate because the rod is "thin".

I haven't actually calculated anything, but the problem might be that in your integral you are making "x" refer to two different things.

In your expression for dm, x is referring to distance from the light end of the rod. But since you are finding moment of inertia and passing the axis through the center of mass, the "x^2" in the integral is referring to distance from the center of mass. The two reference points are not the same.

Maybe if you replaced x^2 by (130/21 - x)^2, it would end up working.

Oh, checked it and it does work. But you have to change the limits to 0 to 10, since I made x the distance from the light end.

Oh yeah... thanks Darksun... :)

## 1. What is moment of inertia?

Moment of inertia is a physical property of an object that describes its resistance to rotational motion. It is the measure of an object's distribution of mass around an axis.

## 2. How is moment of inertia calculated?

The moment of inertia of a rod with varying density can be calculated by integrating the product of the density and the distance from the axis of rotation squared over the entire length of the rod. It can also be calculated using the parallel axis theorem if the moment of inertia of a uniform density rod is known.

## 3. How does density affect the moment of inertia of a rod?

The moment of inertia of a rod is directly proportional to the density of the material. This means that a rod with a higher density will have a higher moment of inertia, making it more difficult to rotate.

## 4. Can moment of inertia be negative?

No, moment of inertia cannot be negative. It is always a positive value as it represents the rotational inertia of an object.

## 5. How does moment of inertia affect the rotational motion of a rod?

The moment of inertia of a rod affects its rotational motion by determining how much torque is required to change its angular velocity. A higher moment of inertia means more torque is needed to accelerate or decelerate the rod's rotation, while a lower moment of inertia means less torque is needed.

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