# Moment of inertia of a sphere

• Za Kh
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.

#### Za Kh

I know the moment of inertia for both a solid sphere and a hollow sphere is , but my teacher has derived a moment of inertia of the sphere but am not sure about what axis she was deriving it , and she got this answer 3/5 MR^2

Za Kh said:
I know the moment of inertia for both a solid sphere and a hollow sphere is , but my teacher has derived a moment of inertia of the sphere but am not sure about what axis she was deriving it , and she got this answer 3/5 MR^2
I also think that there must be a mistake in her derivation. Do you have it? Do you agree with all the steps?

Replusz said:
Was she doing this?
http://imgur.com/dc0ZTvG
Oh! It is clearly wrong. I hope a teacher did not do that in a class!

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

Now I understand , it has turned out that she meant the moment of inertia of the center of sphere and I thought it were the moment of an axis passing through the center , just because she hasn't specified clearly what she meant , thank you everyone :)

How could we calculate the moment of inertia of a sphere , cut into half by the xoy plane ,, its supposed that moment of ineria about the xy and zy axes is zero , i want to know why

Za Kh said:
Now I understand , it has turned out that she meant the moment of inertia of the center of sphere and I thought it were the moment of an axis passing through the center , just because she hasn't specified clearly what she meant , thank you everyone :)
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:
How could we calculate the moment of inertia of a sphere , cut into half by the xoy plane
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 ,, its supposed that moment of ineria about the xy and zy axes is zero , i want to know why
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.

I haven't sensed a physical meaning to it yet in my head , but there's a formula that states tge summation of moments of inertia wrt x-axis ,y axis and z axis , respectively is equal to two times the moment of inertia at the origin . I think so .. Unfortunately i don't have enough time to understand this lesson we've taken lately , my exam is after tomorrow , but am sure that she wrote this title "moment of inertia w.r.t a point"

cnh1995 said:
Do you mean a hemisphere? Solid or hollow?
A solid sphere

jbriggs444 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.
They are not given zero to us , they should be proved by calculation , but am not knowing how

Why so complicated? I guess we assume a homogeneous sphere of density ##\rho=3m/(4 \pi R^3)##. The rotation axis is through the center (all other cases can be evaluated with Steiner's law). Take spherical coordinates and the rotation axis around the polar axis. Then we have
$$\Theta=\rho \int_0^{2 \pi} \mathrm{d} \varphi \int_0^{\pi} \mathrm{d} \vartheta \int_0^{R} \mathrm{d} r r^2 \sin \vartheta r^2 \sin^2 \vartheta = 2 \pi \rho \frac{R^5}{5} \int_{-1}^1 \mathrm{d} u (1-u^2) = \frac{8\pi }{15} \rho R^5=\frac{2}{5} m R^2.$$
In the last step, I've substituted ##u=\cos \vartheta##, ##\mathrm{d} u =\mathrm{d} \vartheta \sin \vartheta##, ##\sin^2 \vartheta=1-u^2##.

## 1. What is moment of inertia of a sphere?

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.

## 2. How is moment of inertia of a sphere calculated?

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.

## 3. What factors affect the moment of inertia of a 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.

## 4. Why is moment of inertia important in physics?

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.

## 5. How does the moment of inertia of a sphere compare to other shapes?

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.