Help understanding torques and moments

[doktorwho]

Homework Statement

Find the period of small oscillation of this system.

Homework Equations

3. The Attempt at a Solution

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I understand the concept of moment of inertia but can't comprehend the first equaion here. The other two i get and they are moments of inertia of other two bodies. (the left and right)

This is supposed to be the total moment of inertia of the system, right?
Now what's left out from the pictures is that the tensions are:
$T_1=mg-maR$ and $T_2=mg+maR$
and so its continued like this:

The moment is $M=F*R$ but i fail to see what the first term represents and what's with the signs?
The last part is:

 

[haruspex]

I do not understand the diagram. Is there a text description of it?
Where is the axis, and in what direction (e.g. normal to the page)?
The two rectangular blocks look like drums seen side on, with strings wrapped around them, with weights suspended. If so, I presume the weights oppose each other.

 

[doktorwho]

I do not understand the diagram. Is there a text description of it?
Where is the axis, and in what direction (e.g. normal to the page)?
The two rectangular blocks look like drums seen side on, with strings wrapped around them, with weights suspended. If so, I presume the weights oppose each other.

Yes i think your correct, the weghts oppose each other as the string is rolled on the other way on one of the drums. They oscilate when one is pulled.

 

[haruspex]

Yes i think your correct, the weghts oppose each other as the string is rolled on the other way on one of the drums. They oscilate when one is pulled.

That suggests the axis is the horizontal bar, but in that case I see no reason for oscillations. If you set it rotating around that bar then it should just keep going at the same rate. All the forces balance.
It looks like there must be some oscillation about an axis normal to the page, probably through the centre of the circle (disc, cylinder, sphere?) or through its highest or lowest point. Depends partly on whether the horizontal bar is is just resting on the circle or fixed to it. Since we are given the mass of it, I guess it is about the centre.

Looking at the first three equations, it seems that the discs at the side are rotating about their centres, so the horizontal bar is the axis. The only way I can then make sense of the I_p term is to say that the central circle has been drawn wrongly. It should be another disc viewed side on..
Summing these as the total moment of inertia about that axis is also wrong. Each suspended mass should contribute mR².

Is there no text with this diagram?

 

[haruspex]

I see no reason for oscillations.

... there is if we take the central circle as a body rigidly attached to the bar, so it swings like a pendulum.

Continuing that thought...
If the circle indeed is a disc as shown, swinging under the axis, its moment of inertia is mR²+mR²/4.