Master Moment of Inertia with Integration: Rod, Disk & Center Axis Solutions

In summary, the conversation discusses problems involving moment of inertia and using integration to find the solution. The first problem involves a rod with a given mass per unit length, while the second problem involves a disk with different surface density functions. The conversation also touches on the importance of using the absolute value when calculating density and the use of polar coordinates to solve the second problem. The expert suggests using the correct expressions and using polar coordinates to solve the problem successfully.
  • #1
acgold
16
0
I've been doing moment of inertia problems and haven't had any trouble so far until these:

Using integration, find the moment of inertia of a rod about an axis through its center if the mass per unit length is [tex]\lambda=\lambda_o x[/tex]. The answer is supposed to come out to [tex]I=\frac 1 8 ML^2[/tex].

Using integration, find the moment of inertia of a disk of radius [tex] a [/tex], about its center if a) [tex]\sigma=\sigma_o r^{-1} [/tex] b) [tex]\sigma=\sigma_o r^2[/tex]. The answers are a) [tex]I=\frac 1 3 Ma^2[/tex] b) [tex]I=\frac 2 3 Ma^2 [/tex]

Even with the answer I'm confused :confused:. Please please help me out...thank you.
 
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  • #2
Okay, I'll do the first one for you.

The line density is given as [tex]\lambda=\lambda_{0}x[/tex]

It's no wonder that you are stuck with this exercise, because this expression is by itself meaningless.
The correct expression must be:
[tex]\lambda=\lambda_{0}|x|[/tex], -L/2<=x<=L/2
i.e the absolute value of x, rather than x itself.
(I assume you gave us ALL the information present in the exercise?)
We gain:
[tex]I=\int_{-\frac{L}{2}}^{\frac{L}{2}}\lambda_{0}|x|{x}^{2}dx=\lambda_{0}\frac{2}{4}(\frac{L}{2})^{4}[/tex]

The mass M of the rod is readily calculated:
[tex]M=\int_{-\frac{L}{2}}^{\frac{L}{2}}\lambda_{0}|x|dx=\lambda_{0}(\frac{L}{2})^{2}[/tex]
Combining the expressions yiels the desired result.
 
  • #3
Thank you thank you thank you...I've been working on that problem for hours. Yeah I gave you ALL the information I had. My professor is notorious for having typos. I did the first problem the way you showed me and I understand it 100% now. I'm off to work on the last two on my own...thanks for all your help. BTW, how did you know it was the absolute value of x? Do the sigma expressions require absolute values also? Thanks again.
 
  • #4
My reasoning was as follows:
1. Density has to be non-negative.
Hence, the only meaningful interpretation of the given expression as it stood, was that x is measured from one end of the rod (x=0 at one end) , but that does not give the stated expression.

2. Hence, I assumed that the density functionshould bemeasured from the point of the rotation axis, but then I had to introduce the absolute value in order to have a meaningful density function.

3. As to the area densities, I don't think so, since the radius pr. definition is non-negative (I haven't checked them yet)
 
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  • #5
Thanks, you're a much better teacher than my professor. :) I've been working on the second problem for a few minutes now and I'm stuck again. Maybe another typo...I don't know but I'll keep trying it.
 
  • #6
Well, thanks for the compliment, :smile:
(But there has to be a good reason why he's a professor, and I'm not!)
Your professor has the second problem right.
In order to solve it, you must use polar coordinates!
Remember that the proper area element, dA, has the form: [tex]dA=rdrd\theta[/tex],
with the limits:
[tex]0\leq{r}\leq{a},0\leq\theta\leq2\pi[/tex]
Good luck!
 

FAQ: Master Moment of Inertia with Integration: Rod, Disk & Center Axis Solutions

What is the "Master Moment of Inertia with Integration" concept?

The "Master Moment of Inertia with Integration" concept is a method used in physics and engineering to calculate the moment of inertia of an object. It involves using calculus and integration to determine the distribution of mass within an object and how it affects its rotational motion.

Why is understanding moment of inertia important?

Understanding moment of inertia is important because it helps us analyze the rotational motion of objects. It is a crucial factor in determining how an object will behave when subjected to a torque, and it also aids in the design and analysis of various mechanical systems.

How is moment of inertia calculated using integration?

To calculate moment of inertia using integration, we divide the object into infinitesimal elements, determine the mass and distance of each element from the axis of rotation, and then integrate the sum of these elements to obtain the total moment of inertia. This method is often used for objects with irregular shapes or varying densities.

What is the difference between rod, disk, and center axis solutions?

Rod, disk, and center axis solutions refer to specific methods of calculating moment of inertia for different types of objects. A rod solution is used for long, thin objects with a constant cross-sectional area, while a disk solution is used for objects with a circular or cylindrical shape. The center axis solution is used for objects with a known axis of symmetry.

How can knowledge of moment of inertia be applied in real-world scenarios?

Moment of inertia is an important concept in many fields, including engineering, physics, and sports. It is used in the design of structures and machines, such as bridges and cars, to ensure they can withstand rotational forces. In sports, it is used to analyze the performance of athletes and equipment, such as in figure skating and gymnastics.

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