Moment of inertia of a cone

In summary, the moment of inertia of a solid cone about its longitudinal axis can be found using the integral I_z = \int\int\int_T(x^2+y^2)dxdydz in cylindrical coordinates. However, the correct moment arm for the second moment from the z axis is ##r^2##, not ##z^2##. Evaluating the integral yields \pi h^5/10, which is the correct answer according to the book.
  • #1
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Homework Statement


Find the moment of inertia of a solid cone about its longitudinal axis (z-axis)

The cone: [itex]x^2+y^2<=z^2, 0<=z<=h[/itex]

[itex]I_z = \int\int\int_T(x^2+y^2)dxdydz[/itex]

Homework Equations


Representing the cone in cylindrical coords:

[itex] x=zcos\theta [/itex]
[itex] y=zsin\theta [/itex]
[itex] z=z [/itex]

The Attempt at a Solution


The integral in cylindrical coords is:

[itex]I_z = \int_0^h \int_0^{2\pi} \int_0^z (z^2)rdrd\theta dz[/itex]

Evaluating the triple integral gives:
[itex]\pi h^5/5[/itex]

But the answer in the book is:
[itex]\pi h^5/10[/itex]

I don't see what I did wrong
 
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  • #2
c0der said:

Homework Statement


Find the moment of inertia of a solid cone about its longitudinal axis (z-axis)

The cone: [itex]x^2+y^2<=z^2, 0<=z<=h[/itex]

[itex]I_z = \int\int\int_T(x^2+y^2)dxdydz[/itex]

Homework Equations


Representing the cone in cylindrical coords:

[itex] x=zcos\theta [/itex]
[itex] y=zsin\theta [/itex]
[itex] z=z [/itex]

The Attempt at a Solution


The integral in cylindrical coords is:

[itex]I_z = \int_0^h \int_0^{2\pi} \int_0^z (z^2)rdrd\theta dz[/itex]

The moment arm for the second moment from the z axis is ##r^2##, not ##z^2##.

Evaluating the triple integral gives:
[itex]\pi h^5/5[/itex]

But the answer in the book is:
[itex]\pi h^5/10[/itex]

I don't see what I did wrong
 
  • #3
I see:

[itex] x=rcos\theta, y=rsin\theta[/itex] where [itex]0<=r<=z[/itex]

Thanks alot
 

What is the formula for calculating the moment of inertia of a cone?

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

How does the moment of inertia of a cone compare to that of a cylinder?

The moment of inertia of a cone is smaller than that of a cylinder with the same mass and radius. This is because the mass of a cone is more concentrated towards the apex, resulting in a smaller moment of inertia.

Does the height of a cone affect its moment of inertia?

Yes, the moment of inertia of a cone is directly proportional to its height. This means that a taller cone will have a larger moment of inertia compared to a shorter cone with the same mass and base radius.

What is the significance of knowing the moment of inertia of a cone?

The moment of inertia of a cone is an important property in physics and engineering. It is commonly used in calculations involving rotational motion and can help determine the stability and strength of structures such as cones used in construction or machinery.

Can the moment of inertia of a cone be negative?

No, the moment of inertia of a cone cannot be negative as it is a measure of an object's resistance to rotational motion. It is always a positive value or zero if the object has no mass or is a point mass.

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