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The general form of moment of inertia does not involve any fractions. However, for examples such as a rod, circular ring etc. has some fraction over it. How is this possible, when the distance between the particle and that of the axis of rotation is R?

Also, what do you mean by "moment of inertia about any axis, parallel from the centre of mass"? If their values are different, are we then assuming that in the case of centre of mass, we consider the rotation of all particles symmetrically placed around it? (Like a circular disc) And the moment of inertia about an axis to be that followed strongly by only a few particles?

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closed as unclear what you're asking by ACuriousMind, Rob Jeffries, user36790, heather, sammy gerbil Aug 20 '16 at 15:21

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  • $\begingroup$ Would you add the example you mention? $\endgroup$ – Steeven Aug 20 '16 at 13:39
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First, be reminded that the moment of inertia is a measure of the resistance of a body to rotation. It is the rotational analog to mass in F=ma. Greater moments require greater force to achieve the same rotational velocity. The moment of inertia of a mass, m, rotating at radius R, is mR^2.

Grab a plastic notebook ruler, the kind with holes in it designed to be stored in a ring binder. Insert a pencil in the center hole and spin it like a helicopter propeller (horizontal plane). It has the moment of inertia of a rod rotating about its center of mass. Now place the pencil in a hole near the end of the ruler and spin it. On my ruler, the axis of rotation is 5.5 inches (14 cm) from the center of mass. The moment of inertia is different. In fact it is greater, as you can feel by the greater force it takes to spin the ruler. Repeat the experiment at different distances, and you will find that the moment of inertia is greater at greater distances from the center of mass.

So what's going on? The spinning ruler is actually a series of particles all rotating at different distances from the axis of rotation (different radii). Each particle has a moment of inertia dependent on its mass and radius. To simplify, we will assume all the particles are identical and have the same mass. Particles close to the axis have smaller radii and lesser moments of inertia, while particles with greater radii have greater moments.

The moment of inertia of the ruler is the sum of all those individual moments. We could divide the ruler into a thousand particles and calculate the moment by adding the individual moments together. We could divide it into a million parts and get a better result. A billion would be even better. A continuous sum would give us an exact value, which is what we do when we perform integration in calculus. An integral is a continuous sum, which is why the integral sign looks like an elongated S?

Why are there fractions? The ruler has a length of 12 inches, the center of mass is at 6 inches, we are spinning it around the center of mass. There are particles rotating at different radii, for instance, 1/16 in, 1/2 in, 1 in, etc. Each of these radii are a fraction of the length of the ruler (12 inches) and result in moments that have fractions due to the fractional lengths. For example, a particle at R/12 results in mR^2/144, while a particle at R/10 results in mR^2/100, etc.

So what does the center of mass have to do with it? The center of mass of a body is the balance point. You can hang a body from its center of mass and it will be in equilibrium, i.e., it will not rotate due to gravity. Because the ruler is uniform, it balances at it physical center, 6 inches. This is the center of mass. It is the point around which the body has rotational equilibrium, those are fancy words meaning the ruler will balance at that point. In addition, the ruler will rotate around the center of mass if you throw it. Even more, it is the center of mass of a rotating projectile that travels with a parabolic path. We treat the body as though all of its mass is concentrated at the center of mass. Further, if a body is suspended from a point other than the center of mass it will always hang so that the center of mass is directly below the suspension point.

Experiment time

1) Cut a piece of thick cardboard into some odd shape. Tie some paper clips to a short length of string. Stick a pin through the cardboard near any edge, then tie the string to the pin. Hold the pin and allow the string and cardboard to hang below it. Take a pen or pencil and trace the line of the string on the cardboard. Repeat the whole process two more times from two different points on the cardboard. The three lines should intersect at one point, the center of mass.

2) Mark that point with a large black dot. Draw a few concentric circles around it, coins do a great job. Hold the cardboard at any point so the flat side is vertical, and toss it upward like tossing a Frisbee on edge vertically. Make sure it spins. You should see the cardboard spinning around the center of mass. Now repeat the toss, but toss it away from you at an angle. the center of mass will follow a parabolic trajectory.

3) Stick the pin in the center of mass and spin the cardboard. Now move the pin to at least three other points and repeat. It is more difficult to spin the cardboard as the pin gets further from the center of mass.

The moment of inertia of a uniform rod rotating about its center is 1/12 mL^2, while the moment around the end of the rod is 1/3 mL^2. The moment around the end of the rod is four times the moment at the center. Spin your ruler again, first with the axis at the center, then at the point near one end, and you will experience the difference.

Summary

1) The center of mass of a body is the point at which the body has rotational equilibrium, i.e., the balance point. The center of mass of a body depends on the shape, i.e, the mass distribution.

2) The moment of inertia of a body is the sum of the moments of inertia of all particles that make up the body, each one having a moment equal to mr^2. Since r for each particle is a fraction of R for the body, the individual moments will be fractions and their sum will also be fractional.

3) The moment of inertia of a body depends on the mass and the distance of the axis of rotation from the center of mass. The greater the distance of the axis of rotation from the center of mass, the greater the moment of inertia.

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The general case for moment of inertia is

$$ I = \sum_i^N m_i r_i^2 \rightarrow \lim_{N \rightarrow \infty} \sum_i^N m_i r_i^2 = \int_Q r^2 dm$$

where $m$ is the mass of the object, $r$ is the distance from the axis of rotation, and $Q$ is the entire mass of the object. It is a summation/integral because you have to consider all of the $N$ point particles ($N \rightarrow \infty)$ that comprise the entire mass $Q$. You can use this formula to derive many specific cases of moment of inertia, such as for a cylinder, sphere, or prism.

Though I suspect you are looking for a more intuitive/conceptual response than a mathematical one. The important part to remember is that moment of inertia depends on the distribution of mass of an object, as well as the center of mass. Therefore it is possible to have different formulas for moment of inertia dependent upon geometry, and specifically, why certain geometries' moments of inertia are fractions of $MR^2$.

I think the best, basic example of this principle is the difference in moment of inertia for a hollow cylinder ($I=MR^2$) and a solid cylinder ($I=MR^2/2$). There is a great derivation here that you can use to follow mathematically, if you choose. Also, you can watch this experimental demonstration.

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