Physic I

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The moment of inertia of a given distribution of mass with respect to an axis of rotation is by DEFINITION

`I =int R^2*dm`    (1)

where the integral is taken for all values of distance `R` measured from the axis of rotation to the infinitesimal mass `dm` . The case of a plane (x,y) distribution of mass is given in the figure below (the axis of rotation is perpendicular to the figure):

`I_z = int (x^2+y^2)*dm`

This means that the moment of inertia is maximum when the mass is distributed at higher distances from the rotation axis.

The highest moment of inertia will have the flat hoop, since for it all mass is distributed at the highest distance `R = R_(max)` from the center.

Now, between the sphere (uniform) and the upright hoop both have the mass distributed uniform from `R=0` to `R=R_max` , but the sphere has a symmetrical distribution over all three x,y,z coordinates, whereas the upright hoop has a symmetrical distribution over only two coordinates. This means that the infinitesimal mass `dm=M/V` will be larger for hoop than for the sphere and the moment of inertia of the sphere (uniform) will be the smallest (see equation (1)).

Thus the flat hoop will have the highest moment of inertia, while the sphere will have the smallest moment of inertia.

Indeed, the computations give:

For the flat hoop `I =MR^2`

` ` For the upwright hoop `I =(MR^2)/2`

For the sphere `I =(2MR^2)/5`

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