Moment of Inertia of Spheres/Discs: Adding & Dependency

[kushan]

If we have two coaxial spheres each mass m , and radius r are rotating about an axis ,
what will be the moment of inertia of the system about the axis ,
what will it be if we have two discs ( mass m , radius r ) ?

i think the moment of inertia simply adds , is this right

other thing which is bugging me ? will we add the moment of inertia

in the figure below , all balls are solid and same ,
will the moment of inertia be added here also ?
does it depend on the angle subtended at centre( symmetric system ) , or is it independent of it ?
and what if i just increase the distance of one ball from axis ( increasing its orbit) ?
please help

 

[Filip Larsen]

If we have two coaxial spheres each mass m , and radius r are rotating about an axis ,
what will be the moment of inertia of the system about the axis ,
what will it be if we have two discs ( mass m , radius r ) ?

i think the moment of inertia simply adds , is this right

Yes, that is correct. The moments of inertia for two or more objects can be algebraically added together to form the total moment of inertia if all are referring to the same axis (coaxial, as you said). If they are not referring to the same axis the moments of inertia first have to be transformed to a common axis, for instance by using the parallel axis theorem [1].

other thing which is bugging me ? will we add the moment of inertia

in the figure below , all balls are solid and same ,
will the moment of inertia be added here also ?

It depends on what axis you are referring to. For instance, if the moment of inertia for each sphere is expressed about an axis through the center of the sphere, then you cannot just add them because those axes are not coaxial on your figure.

If you want the total moment of inertia around the center of the orbit, you would first have to transform the moment of inertia for each sphere to this common axis using the parallel axis theorem. If all the spheres have same material properties and all are places at the same distance from this common center you will get three identical contributions to the total moment of inertia around the center. If you place one of the sphere in a different distance it will contribute differently than the other due to the different distance used in the parallel axis theorem.

does it depend on the angle subtended at centre( symmetric system ) , or is it independent of it ?
and what if i just increase the distance of one ball from axis ( increasing its orbit) ?
please help

The contribution of each part to the total moment of inertia around a certain axis does only depend on the distance of this object center of mass to the axis. For example, the total moment of inertia about the vertical center axis for your three spheres will be independent of any rotation of the spheres around this axis, so you would get same moment of inertia with all spheres placed at the same point as when they are distributed evenly along the circle.

If you change the distance from the axis for one of the spheres, the contribution to the total moment of inertia will change, as I mentioned above.

[1] http://en.wikipedia.org/wiki/Parallel_axis_theorem

 

[kushan]

thanks ,
can u like give some mathematical equation for the moment of inertia of balls ( in the figure ) about the axis ,which is shown (all same r radius and R distance from axis in case I
and Different R in case II )

 

[Filip Larsen]

You should be able to easily derive those equations yourself from the moment of inertia around an axis through the center of a sphere (which can be found on the net and in most references and textbooks covering moment of inertia) and from the relation established by the parallel axis theorem.

Feel free to post your work here (relevant equations and calculations) if you get stuck or if you just want to have your result checked.