I'm working on a problem involving deriving equations of rotational motion for a system of connected rods (see the picture), and I'm a bit confused about a few things to do with rotational motion. I won't go into detail about the forces involved or their precise nature - it's more something I don't understand about rotational motion. I know how to derive local and global momentum balances by taking forces acting on each point, applying Newton's second law and also considering the motion of the centre of mass of the whole system, but I am getting incredibly confused with doing the same for the rotational motion.
If the moments of inertia about each of the first three blue dots are $I_1$, $I_2$, $I_3$, would we have the following (using conservation of angular momentum, i.e. equivalent to Newton's Second Law for rotational motion):
- $\frac{\mathrm{d}}{\mathrm{d}t}(I_1 \dot{\alpha})=$ net anticlockwise moment on rod 1 about the first point,
- $\frac{\mathrm{d}}{\mathrm{d}t}(I_2 \dot{\beta})=$ net anticlockwise moment on rod 2 about the second point,
- $\frac{\mathrm{d}}{\mathrm{d}t}(I_3 \dot{\gamma})=$ net anticlockwise moment on rod 3 about the third point?
Also, how would we write down the equation for the overall moment balance? One of my supervisors suggested that the global moment balance can be obtained by considering
$$\frac{\mathrm{d}}{\mathrm{d}t}(I_1 \dot{\alpha})=\sum\textrm{(anticlockwise moments over system as a whole)} $$
since $\alpha$ could be describing the orientation of the whole body as the rods are connected, but I'm not so sure about this. Surely it doesn't make sense for the LHS of the above to equal two different things?
Also, a few of questions:
- Is it usually true that adding all the angular momentum equations for each rod (constituent parts of a system) would result in the overall angular momentum balance equation, just like you can close a system of N2L equations by adding them all together and recovering an equation for the whole system?
- Would you still be able to write down an equation of angular momentum of the first rod involving $\alpha$ if moments on the rod are taken to be about a different point on it? I know the moment of inertia would be different, but what about the angle?
- Is $\alpha$ an appropriate choice of angle for describing the orientation of the whole system here?