If we have five identical rigid rods, each of length l and mass m, are connected together to form the system shown in the figure. The system may rotate about an axis passing through AB.

The question is to find the moment of inertia of the system with respect to axis AB. enter image description here

So what I did considered BC as a rod rotating about its end. I then considered DC as a mass connected to BC (but I doubt this one)

And the rest I do not know how to consider them. I would appreciate some guidance on how to think of moment of inertia in this problem.


1 Answer 1


The missing piece you need is The Parallel Axis Theorem, which is incredibly useful for calculating moments of intertia in general. Let us know if you still have specific issues (e.g. how to implement this).

The simple equations for moments of inertia for individual objects (e.g. $I = \frac{1}{3} m L^2$ for a rod about its end) are specifically with respect to a certain axis of rotation. The difficulty is when you need to know the moment of inertia about a different axis.
For example, the moment of inertia for $DE$ could easily be calculated if it was rotating about the line $CD$ --- it would just be $I_{DE,0} = mL^2/3$. But we want to know what it is about the line $AB$ instead --- so we use the Parallel Axis Theorem, $$I = I_0 + mx^2$$ (for an offset distance $x$, which in this case is the length of the rod $BC$, so $x=L$. Thus, for the moment of Inertia of $DE$, about $AB$ is: $$I_{DE} = mL^2/3 + mL^2 = 4mL^2/3$$
Note that this works because the original axis of rotation ($CD$) is parallel to the new axis of rotation ($AB$).

  • $\begingroup$ Yes please, can you tell me how does the parallel axis theorem comes in handy here? Is it for specific rods or all rods except BC? @zhermes $\endgroup$ Feb 12, 2015 at 16:10
  • $\begingroup$ Aha, I see. You said that this worked because CD was parallel to BC. But what to do with the rest of the rods? Like for example what is the MOI of CD wrt AB? or EF? $\endgroup$ Feb 12, 2015 at 16:33
  • $\begingroup$ What I meant was that the 'original' axis of rotation, has to be parallel to the 'new' axis (i.e. the axis which you want the answer for). You can find a way to do this for each of the rods, e.g. compare $BC$ to $DE$. $\endgroup$ Feb 12, 2015 at 16:37
  • $\begingroup$ But one second, what is moment of inertia of CD wrt AB? Is it really acting like a mass like I said in my question? $\endgroup$ Feb 12, 2015 at 16:50
  • $\begingroup$ Well, how can you calculate that using the same principles? $\endgroup$ Feb 12, 2015 at 18:49

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