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- Thread starter Za Kh
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In summary, the conversation discusses the moment of inertia for a solid sphere and a hollow sphere, with one person questioning their teacher's derivation of a moment of inertia for the sphere. They also mention Steiner's law and a similar problem posted on a forum. The conversation then shifts to calculating the moment of inertia for a sphere cut in half by the xoy plane, with one person asking for clarification on the meaning of a moment of inertia at a point. The conversation concludes with a discussion on calculating the moment of inertia for a solid sphere and a suggestion to use spherical coordinates.

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https://en.wikipedia.org/wiki/Parallel_axis_theorem

Although i would rather think there was a mistake.

Someone seemed to have a similar problem: https://www.physicsforums.com/threads/moment-of-inertia-for-a-rectangular-plate.12903/

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I also think that there must be a mistake in her derivation. Do you have it? Do you agree with all the steps?Za Kh said:

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Was she doing this?

http://imgur.com/dc0ZTvG

http://imgur.com/dc0ZTvG

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Oh! It is clearly wrong. I hope a teacher did not do that in a class!Replusz said:Was she doing this?

http://imgur.com/dc0ZTvG

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I want to show you the paper on my notebook but I don't know how to send a pic here

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But what does that mean exactly? There is no way to make a sphere rotate in such a way that it will have that moment of inertia, so it is actually a completely unphysical result. That's why it is never quoted as a moment of inertia of a sphere.Za Kh said:

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Do you mean a hemisphere? Solid or hollow?Za Kh said:How could we calculate the moment of inertia of a sphere , cut into half by the xoy plane

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The "xy" axis is a synonym for an axis in the z direction? So if the moment of inertia about this axis is zero, every point within the sphere must be somewhere on the z axis.Za Kh said:

The "zy" axis is a synonym for an axis in the x direction? So if the moment of inertia about this axis is zero, every point within the sphere must be somewhere on the x axis.

It follows that every point within the sphere must be at the intersection of the x and z axes. i.e. at the origin. Well, yeah, that moment of inertia is fairly easy to calculate.

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A solid spherecnh1995 said:Do you mean a hemisphere? Solid or hollow?

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They are not given zero to us , they should be proved by calculation , but am not knowing howjbriggs444 said:The "xy" axis is a synonym for an axis in the z direction? So if the moment of inertia about this axis is zero, every point within the sphere must be somewhere on the z axis.

The "zy" axis is a synonym for an axis in the x direction? So if the moment of inertia about this axis is zero, every point within the sphere must be somewhere on the x axis.

It follows that every point within the sphere must be at the intersection of the x and z axes. i.e. at the origin. Well, yeah, that moment of inertia is fairly easy to calculate.

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$$\Theta=\rho \int_0^{2 \pi} \mathrm{d} \varphi \int_0^{\pi} \mathrm{d} \vartheta \int_0^{R} \mathrm{d} r r^2 \sin \vartheta r^2 \sin^2 \vartheta = 2 \pi \rho \frac{R^5}{5} \int_{-1}^1 \mathrm{d} u (1-u^2) = \frac{8\pi }{15} \rho R^5=\frac{2}{5} m R^2.$$

In the last step, I've substituted ##u=\cos \vartheta##, ##\mathrm{d} u =\mathrm{d} \vartheta \sin \vartheta##, ##\sin^2 \vartheta=1-u^2##.

The moment of inertia of a sphere is a measure of its resistance to rotational motion. It is the rotational analog of mass in linear motion, and is determined by the distribution of mass within the sphere and its distance from the axis of rotation.

The moment of inertia of a sphere can be calculated using the formula I = (2/5)mr^2, where m is the mass of the sphere and r is the radius. This formula is derived from the integration of infinitesimal elements of mass within the sphere.

The moment of inertia of a sphere is affected by its mass and radius, as well as the distribution of mass within the sphere. A larger mass or radius will result in a larger moment of inertia, while a more spread out mass distribution will decrease the moment of inertia.

Moment of inertia plays a crucial role in rotational dynamics, as it determines how much torque is required to accelerate a sphere into rotational motion. It is also used in the calculation of angular momentum and the conservation of angular momentum in various physical systems.

The moment of inertia of a sphere is the lowest of all solid shapes with the same mass and radius. This means that a sphere requires the least amount of torque to start rotating compared to other shapes. For example, a solid cylinder with the same mass and radius as a sphere would have a moment of inertia that is 1.5 times greater.

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