Moment of inertia of a solid of rotation

• Grand
In summary, the conversation revolves around finding the integral that represents the moment of inertia of a solid of rotation created by rotating the first quadrant area bounded by the curve x^3+y^3=8 around the y-axis. The proposed answer is I_y=M\frac{\int_0^2x^2ydx}{\int_0^2x^2dx}, but it is deemed incorrect. The correct approach involves using cylindrical shells instead of disks, as well as calculating the moment of inertia of a cylindrical shell as 1/2 ρr4dy.
Grand

Homework Statement

The first quadrant area bounded by the curve
$$x^3+y^3=8$$
is rotated around y-axis to give a solid of rotation. The question asks for an integral which represents the solid's moment of inertia around the axis.

$$I_y=M\frac{\int_0^2x^2ydx}{\int_0^2x^2dx}$$

but seems like I'm wrong. Could someone help me out, please?

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The Moment of Inertial of a cylindrical shell of mass, m, and radius, R, is: m(R2).

We need to integrate over the height, that is from 0 to 2, and for each y we need to account for a disk of radius
$$x=\sqrt[3]{(8-y^3)}$$
which moment of inertia is:
$$dI=\rho x^2 dy=\rho (8-y^3)^{2/3}dy$$
For the whole body this is:
$$I=\int_0^2\rho (8-y^3)^{2/3} dy$$
$$I=\frac{m}{V}\int_0^2 (8-y^3)^{2/3} dy$$

I think this is wrong.

The Moment of Inertial of a disk of mass, m, and radius, R, is: (½)m(R2).

However, I think it would be better to use cylindrical shells as tiny-tim recommended.

The last integral you give, $\displaystyle \int_0^2 (8-y^3)^{2/3} dy$ , gives the volume of your solid of revolution.

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Hi Grand!
Grand said:
We need to integrate over the height, that is from 0 to 2, and for each y we need to account for a disk of radius
$$x=\sqrt[3]{(8-y^3)}$$
which moment of inertia is:
$$dI=\rho x^2 dy=\rho (8-y^3)^{2/3}dy$$

This is wrong for two reasons:

i] the moment of inertia of a disc is 1/2 mr2, not mr2

ii] you can't just use ρ instead of m … you need m for the whole disc, which is ρr2dy, so the whole formula for moment of inertia of the disc is 1/2 ρr4dy

But why are you making things so difficult for yourself? As SammyS says, cylindrical shells would be much easier.

Try again.

1. What is the definition of moment of inertia of a solid of rotation?

The moment of inertia of a solid of rotation is a measure of its resistance to rotational motion. It is a physical property that depends on the shape, size, and mass distribution of the object.

2. How is the moment of inertia calculated for a solid of rotation?

The moment of inertia for a solid of rotation can be calculated using the formula I = ∫r²dm, where r is the distance from the axis of rotation to the mass element dm. This integral is taken over the entire mass of the object.

3. What factors affect the moment of inertia of a solid of rotation?

The moment of inertia of a solid of rotation is affected by the mass, shape, and distribution of mass in the object. Objects with more mass, or with the mass distributed further from the axis of rotation, tend to have a larger moment of inertia.

4. How does the moment of inertia affect the rotational motion of a solid of rotation?

The moment of inertia plays a crucial role in determining the rotational motion of a solid of rotation. Objects with a larger moment of inertia require more torque to rotate and will have a slower rotational speed compared to objects with a smaller moment of inertia.

5. Can the moment of inertia of a solid of rotation be changed?

Yes, the moment of inertia of a solid of rotation can be changed by altering its shape or mass distribution. This is why objects like figure skaters can change their moment of inertia by changing their body position, allowing them to spin faster or slower.

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