Moments of Inertia by Integration

In summary, the authors derived a new formula for moments of inertia which is based off of the first formula but uses the height of the rectangle instead of the width.
  • #1
CivilSigma
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Homework Statement


I am having trouble understanding the formula for Moments Of Inertia by direct integration.

Homework Equations


I understand the following (which is the definition) :

$$ I_x = \int y^2 dA $$ $$I_y = \int x^2 dA $$

However come to application on a problem, my book doesn't even use those formulas.

The authors derived a new formula:

$$ dI_x =\frac{1}{3} y^3 dx $$ $$I_x = \int dI_x$$

Here is a screen shot of what they did:

http://tinypic.com/r/2vjr5vq/8

Why do they do that? Why not just use the definition and solve?

The Attempt at a Solution

 
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  • #2
The first formula for ##I_x## is a summation over horizontal strips between ##y## and ##y+dy##. The integration is over ##y## so in the integrand ##y## is the integration variable. The strips are not assumed to be all the same width, so the shape being integrated need not be a rectangle, and need not touch the ##x## axis.

The second formula for ##I_x## is a summation over vertical strips between ##x## and ##x+dx##. The integration is over ##x##. In the integrand ##y## represents the height of the top of the vertical strip above the ##x## axis. It assumes that the base of the strip is the ##x## axis.

The second formula is derived from the first, by using the formula for MOI of a vertical rectangle that was just derived.

The alternative formulation is provided because sometimes it will be easier to integrate over ##x## than over ##y##. In any given situation, one can choose the method that gives the easiest calculation. Note however that the first method is more general, as the second can only be used when the base of the shape is flat and runs along the ##x## axis, and the shape has no parts that project sideways beyond the base.
 
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  • #3
Thank you for the reply andrewkirk!

I have to understand the use of $dI_x$ in the formula because we are summing up the moments of inertia of all the rectangles.

So, I'm just curious, how would I go about solving problems given the original (definition) equation for moments inertia?
 
  • #4
sakonpure6 said:
how would I go about solving problems given the original (definition) equation for moments inertia?
You use the first formula you've written above, and you replace ##dA## by ##w(y)dy## where ##w(y)## is the width of the shape at height ##y##.
 
  • #5
For both $I_x$ and $I_y$ ?
 
  • #6
No, just for ##I_x##. To do ##I_y## you swap ##x## and ##y## in all formulas.

By the way, the code to get in-line latex math symbols on physicsforums is two consecutive # characters, not a single $. That's an annoying difference with Texmaker, which is what I use elsewhere.
 
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  • #7
Thank you Andre! I'll try some problems and hopefully I get them right.
 

1. What is the concept of "Moments of Inertia by Integration"?

The concept of "Moments of Inertia by Integration" is a mathematical method used to calculate the rotational inertia of an object. It involves integrating the mass of an object over its entire volume or area, taking into account the distribution of mass and distance from the axis of rotation.

2. How is the moment of inertia calculated using integration?

The moment of inertia is calculated by finding the integral of the product of the mass and the square of the distance from the axis of rotation. This integral is evaluated over the entire volume or area of the object, depending on its shape.

3. What is the importance of knowing the moment of inertia of an object?

Knowing the moment of inertia of an object is important in understanding its rotational motion and stability. It is also crucial in designing machines and structures that require precise control over their rotational motion.

4. How does the distribution of mass affect the moment of inertia?

The distribution of mass greatly affects the moment of inertia as it determines the distance of each mass element from the axis of rotation. Objects with a higher concentration of mass near the axis will have a lower moment of inertia compared to those with a more spread out distribution of mass.

5. Can the moment of inertia be negative?

No, the moment of inertia cannot be negative as it is a measure of an object's resistance to change in rotational motion. A negative moment of inertia would imply that an object is resisting rotation in the opposite direction, which is not possible.

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