Calculating Polar Moment of a Region Inside/Outside Circle and Cardiod

In summary, the problem is asking to find the polar moment of a region inside a circle with radius 3 and outside a cardioid with radius 2 + sin(theta). The polar moment is represented by I0 and can be calculated by integrating the function r^3 * theta over the given limits. The setup for the integral appears to be correct and no manual evaluation is needed. The cardioid and circle only intersect at one point.
  • #1
VinnyCee
489
0
Here is the problem:

Find the polar moment of the region that lies inside the circle [tex]r = 3[/tex] and outside the cardiod [tex]r = 2 + \sin\theta[/tex]. Assume [tex]\delta = r\theta[/tex]

Here is what I have:

[tex]I_{0} = I_{x} + I_{y}[/tex]

[tex]I_{0} = \int_{0}^{2\pi}\int_{2 + \sin\theta}^{3}\;r^3\;\theta\;\sin^2\theta\;dr\;d\theta + \int_{0}^{2\pi}\int_{2 + \sin\theta}^{3}\;r^3\;\theta\;\cos^2\theta\;dr\;d\theta[/tex]

[tex]I_{0} = \int_{0}^{2\pi}\int_{2 + \sin\theta}^{3}\;r^3\;\theta\;dr\;d\theta[/tex]

Is this the correct setup? I don't have to manually evaluate this one, I just need to setup the integral limits and the integrand. Thank you in advance!
 
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  • #2
It looks okay to me...The cardioide & the circle have only one common point (y=3)

Daniel.
 
  • #3


Yes, your setup looks correct. The integral limits and integrand are the most important parts when setting up a polar moment integral. As long as those are correct, you should be able to evaluate the integral using a calculator or computer program. Good job!
 

What is the formula for calculating the polar moment of a region inside a circle?

The formula for calculating the polar moment of a region inside a circle is given by Ip = πr4/4, where r is the radius of the circle.

What is the formula for calculating the polar moment of a region outside a circle?

The formula for calculating the polar moment of a region outside a circle is given by Ip = π(R4 - r4)/4, where R is the radius of the larger circle and r is the radius of the smaller circle.

What is the formula for calculating the polar moment of a region inside a cardioid?

The formula for calculating the polar moment of a region inside a cardioid is given by Ip = 3πr4/20, where r is the radius of the cardioid.

What is the formula for calculating the polar moment of a region outside a cardioid?

The formula for calculating the polar moment of a region outside a cardioid is given by Ip = π(R4 - r4)/5, where R is the radius of the larger circle and r is the radius of the smaller circle.

What are some applications of calculating polar moment of a region?

The calculation of polar moment of a region is commonly used in engineering and physics, specifically in the design and analysis of structures and machines. It helps in determining the resistance to torsion and bending of a given region. It is also useful in calculating the moment of inertia, which is important in understanding the rotational motion of objects.

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