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

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SUMMARY

The forum discussion focuses on calculating the mass, second moment, and radius of gyration for a doughnut-shaped yoyo using double and triple integrals in polar coordinates. The mass is determined using the equation mass = ∫∫_D ρ(x,y) dA, while the radius of gyration is calculated with R = √(I/M), where I represents the moment of inertia and M is the mass. The second moment is clarified as synonymous with the moment of inertia, requiring adjustments in the integral for calculations about the x and y axes. Participants emphasize the complexity of the problem and share insights on approaching the calculations.

PREREQUISITES
  • Understanding of double and triple integrals in polar coordinates
  • Familiarity with the concepts of mass and moment of inertia
  • Knowledge of calculus, specifically Calculus IV topics
  • Ability to apply integrals to physical shapes and objects
NEXT STEPS
  • Study the application of double integrals in polar coordinates for complex shapes
  • Learn about the derivation and application of the moment of inertia
  • Explore the concept of radius of gyration in various physical contexts
  • Practice problems involving mass and moments of inertia for different geometric shapes
USEFUL FOR

Students and educators in advanced calculus, particularly those focusing on applications in physics and engineering, will benefit from this discussion. It is especially relevant for anyone tackling complex integrals and physical properties of objects.

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 x and y axes...

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

R= {sqrt{(I/M)}} , 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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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.
 
I had found that soon after I posted but thanks for replying. Still working on this one. Its a toughie.
 

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