Finding moment of inertia of cone

In summary, when finding the moment of inertia for a cone standing on its vertex, the R limits in the triple integral are integrated as 0 to (R/h)z. This is dependent on the order of integration, where if the integration is done earlier, the maximum value of r is constrained by the current value of z in the outer integral. The R value represents the maximum radius at height h, and the variable r represents the radius at height z. The angle of the cone's slope to vertical can be represented as tan(θ) = R/h = r/z.
  • #1
Vitani11
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Homework Statement


Why is it that when you integrate to find the moment of inertia of a cone standing on its vertex (like a spinning top) with height h mass M and radius R do you integrate the R limits as 0 to (R/h)z in the triple integral (cylindrical coordinates) below?

Homework Equations


I = moment of inertia
D = density (M/πR2h)
ρ = R distance from rotation axis (limits from 0 to (R\h)z)
φ = 2π the angle swept (limits 0 to 2π)
z = h the height of the cone (limits 0 to h)

The Attempt at a Solution


I = ∫ρ2dm = D∫∫∫ρ3dρdφdz
 
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  • #2
I assume you mean the r limits, not R limits.
It depends on the order of integration. If the integration wrt r is the last step then the range is 0 to R. If it is an earlier step then the maximum value of r is constrained by the current value of z in the outer integral.
 
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  • #3
haruspex said:
If it is an earlier step then the maximum value of r is constrained by the current value of z in the outer integral.

Yeah this is the case but I don't understand where it came from. Why doesn't it involve sines? How do you see it is (R/h)z from the picture? For the problem is was given as capital R for radius
 
  • #4
Vitani11 said:
Yeah this is the case but I don't understand where it came from. Why doesn't it involve sines? How do you see it is (R/h)z from the picture? For the problem is was given as capital R for radius
R is the maximum radius, i.e. the radius at height h. For the integral, you need a variable for the radius at height z. r seems a reasonable choice.
If the angle of the cone (slope to vertical) is θ then tan(θ) = R/h = r/z.
 
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  • #5
Okay thanks
 

1. What is the formula for finding the moment of inertia of a cone?

The formula for finding the moment of inertia of a cone is: I = (3/10)mr^2, where m is the mass of the cone and r is the radius of the base.

2. How do I determine the mass of the cone for calculating moment of inertia?

To determine the mass of the cone, you can use a scale or balance to measure its weight. Alternatively, you can also calculate the mass by multiplying the density of the material by the volume of the cone.

3. What is the significance of finding the moment of inertia of a cone?

The moment of inertia of a cone is a measure of its resistance to rotational motion. It is an important parameter in understanding the stability and behavior of the cone when subjected to external forces.

4. Can the moment of inertia of a cone change?

Yes, the moment of inertia of a cone can change if the mass or the radius of the cone changes. It can also change if the cone's axis of rotation is altered.

5. Are there any real-life applications of finding the moment of inertia of a cone?

Yes, the concept of moment of inertia is used in various fields such as engineering, physics, and sports. For example, it is crucial in designing stable structures like bridges and buildings. In sports, it is used to optimize the performance of equipment like golf clubs and tennis rackets.

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