Moment Of Inertia Of Cone

Introduction: Moment of Inertia

The moment of inertia of a rigid object, commonly known as the mass moment of inertia, angular momentum, or rotational inertia, is a quantity that specifies the torque required by a rotational axis for a desired angular acceleration; analogous to how mass calculates the force needed for the desired acceleration.

Consider a solid uniform cone with mass M, radius R, and height h. Then density is

p= \[\frac{M}{\frac{1}{3}πR^2h}\],

And the moment of inertia tensor over the base center is

\[\int_{V}p(x,y,z)\begin{bmatrix} y^2 + z^2 & -xy & -xz \\ -xy & z^2 + x^2 & -yz \\ -xz & -yz & x^2 + y^2\end{bmatrix}dxdydz\]

\[\begin{bmatrix} \frac{1}{10}Mh^2 + \frac{3}{20}MR^2 & 0 & 0 \\ 0 & \frac{1}{10}Mh^2 + \frac{3}{20}MR^2 & 0 \\ 0 & 0 & \frac{3}{10}MR^2 \end{bmatrix}\],

\[p\int_{h}^{0}\int_{-R(h-z)/h}^{R(h-z)/h}\int_{-\sqrt{[R(h-z)/h]^2-y^2}}^{\sqrt{[R(h-z)/h]^2-y^2}}\begin{bmatrix} y^2 + z^2 & -xy & -xz \\ -xy & z^2 + x^2 & -yz \\ -xz & -yz & x^2 + y^2\end{bmatrix}dxdydz\]


Moment of Inertia of Circular Cone Derivation

Here we'll look at the derivation as well as the approximation to find the moment of inertia about an axis of a standard right circular cone.

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We divide the cone into a small elementary disk where we consider the radius of the cone to be r at a distance x. Breakthrough. We'll need to find the disk mass, and it's given as;

dm = M / ⅓ π R2H (πr2dx)

dm = 3 M / π R2H (R2 / H2 X x2) dx

dm = 3Mx2dx / H3

Now if we consider the triangle's similarity we obtain;

r / x = R / H

r = R / H X x

We also need to remember the moment of elementary disk inertia which is;

½ dmr2

Meanwhile,

dI = ½ (3Mx2dx / H3) . R2 / H2 X x2

I = ∫dI = 3/2 MR2 / H5 . H5 / 5 = 3MR2 / 10

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FAQs (Frequently Asked Questions)

1.What is the Moment of Inertia of a Hollow Cone?

Ans. To derive the moment of a hollow cone's inertia formula, we can basically follow some general guidelines that are; calculating the mass density per unit area.

2.How is the Moment of Inertia calculated?

Ans. Basically, the moment of inertia can be determined for any rotating object by taking the distance of each particle from the rotation axis (r in the equation), squaring that value (that is the r2 term) and multiplying it by the mass of that particle.

3. What is the Centre of the Mass of a Hollow Cone?

Ans. The surface of a hollow cone may be considered to consist of an infinite number of infinitesimally slender triangles of isosceles, and thus the center of mass of a hollow cone (without foundation) is 2/3 of the way from the pole to the base midpoint.

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