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

• VinnyCee
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.
VinnyCee
Here is the problem:

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

Here is what I have:

$$I_{0} = I_{x} + I_{y}$$

$$I_{0} = \int_{0}^{2\pi}\int_{2 + \sin\theta}^{3}\;r^3\;\theta\;\sin^2\theta\;dr\;d\theta + \int_{0}^{2\pi}\int_{2 + \sin\theta}^{3}\;r^3\;\theta\;\cos^2\theta\;dr\;d\theta$$

$$I_{0} = \int_{0}^{2\pi}\int_{2 + \sin\theta}^{3}\;r^3\;\theta\;dr\;d\theta$$

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!

It looks okay to me...The cardioide & the circle have only one common point (y=3)

Daniel.

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.

• Introductory Physics Homework Help
Replies
6
Views
1K
• Introductory Physics Homework Help
Replies
13
Views
2K
• Introductory Physics Homework Help
Replies
1
Views
185
• Introductory Physics Homework Help
Replies
4
Views
806
• Introductory Physics Homework Help
Replies
17
Views
372
• Introductory Physics Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
869
• Introductory Physics Homework Help
Replies
16
Views
1K
• Introductory Physics Homework Help
Replies
21
Views
1K