Calculus IV: Calculating Mass, 2nd Moment & Radius of Gyration for Doughnut Yoyo

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In summary, the conversation revolves around a problem in Calculus IV involving double and triple integrals in polar coordinates. The problem asks for the mass, 2nd moment (or moment of inertia), and radius of gyration of a doughnut shaped yoyo about both the x and y axes. The equations given are for finding the mass and radius of gyration, but there is confusion about what the 2nd moment represents. It is clarified that the 2nd moment is indeed the moment of inertia, and the process for finding it is explained. The person asking for help thanks the responder for their input.
  • #1
kylemadigan
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Doughnut Calculus!

Hey guys, I need help with a somewhat big Calculus IV problem. We are reviewing double and triple integrals in polar coordinates. So here is my problem:

Work out the mass, 2nd moment, and radius of gyration for a doughnut shaped yoyo about both [tex]x[/tex] and [tex]y[/tex] axes...

I know the following equations: mass = [tex]\iint_{D} \rho(x,y) dA[/tex]

[tex] R= {sqrt{(I/M)}}[/tex] , where I is the moment of inertia and M is the mass of the lamina (object).
Have no clue about what 2nd moment is? is this like another calculation of moments of inertia or moment about an axis?

Anyways, can someone help me get started on this one please?
 
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  • #2
From what I recall, the 2nd moment, in physics terms, is the moment of inertia. So if you are doing it about the y-axis, you mulitply your term within the integral by x ( or r cos theta), and do the vice-versa when you are doing it about the x - axis.
 
  • #3
I had found that soon after I posted but thanks for replying. Still working on this one. Its a toughie.
 

1. What is Calculus IV and how does it relate to doughnut yoyo?

Calculus IV is a branch of mathematics that deals with the study of rates of change and accumulation. It specifically focuses on the application of calculus principles to calculate the mass, 2nd moment, and radius of gyration for a doughnut yoyo. This involves using techniques such as integration, differentiation, and volume calculations.

2. Why is it important to calculate the mass, 2nd moment, and radius of gyration for a doughnut yoyo?

These calculations help us understand the physical properties and behavior of the doughnut yoyo. The mass and 2nd moment provide information about its weight distribution and rotational inertia, which affect its stability and movement. The radius of gyration is also crucial in determining the yoyo's moment of inertia, which is important in understanding its angular acceleration.

3. What is the process for calculating the mass, 2nd moment, and radius of gyration for a doughnut yoyo?

The first step is to determine the mass of the yoyo by measuring its weight. Then, using the dimensions of the yoyo, we can calculate its volume and density. From there, we can use integration to find the 2nd moment of the yoyo and the radius of gyration. These calculations involve using the formula for moments of inertia and integrating over the volume of the yoyo.

4. Are there any real-life applications for these calculations?

Yes, these calculations have practical applications in many fields such as physics, engineering, and sports. Understanding the mass, 2nd moment, and radius of gyration for a doughnut yoyo can help in designing more efficient and stable yoyos for competitive play. It can also be used in the design and analysis of other objects with rotational motion, such as gears and propellers.

5. Is Calculus IV the only way to calculate these properties for a doughnut yoyo?

No, there are other mathematical techniques that can be used to calculate these properties, such as vector calculus and linear algebra. However, Calculus IV is specifically designed to handle complex and variable shapes, making it a suitable tool for calculating mass, 2nd moment, and radius of gyration for a doughnut yoyo.

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