# Does a general rotational analogue of CoM work exist?

For a general body, there exists the notion of centre of mass work; that is, computing the work done if all of the forces on the body act through the centre of mass. If we separate the total kinetic energy into that of the CM and that relative to the CM, ##T = T_{CM} + T*##, we can show by integrating Newton's second law that if ##\vec{F}## is the resultant force, $$\int \vec{F} \cdot d\vec{r}_{CM} = \Delta T_{CM}$$Evidently since ##\Delta T = \Delta T_{CM} + \Delta T*##, it would be nice if we could also find an expression for ##\Delta T*##.

For rigid bodies, such an expression is not hard to find. If, in one dimension, ##\tau## is the total torque about the centre of mass, then we can integrate w.r.t. the angular position of the rigid body $$\int_{\theta_1}^{\theta_2} \tau d\theta = \int_{\theta_1}^{\theta_2} I_{CM} \frac{d\omega}{dt} d\theta = \Delta \left(\frac{1}{2} I_{CM} \omega^{2}\right) = \Delta T*$$ That's nice, but the general equation for CM-work ##\int \vec{F} \cdot d\vec{r}_{CM} = \Delta T_{CM}## also applies to bodies that are not rigid. But with the rotational version above, it is required that we know the angular displacement of the rigid body ##\theta_2 - \theta_1## and consequently we cannot use it for non-rigid bodies which do not have a well defined angular displacement.

So I wondered whether anyone knew of a form with which we can calculate ##\Delta T*##, without computing the real work ##\Delta T##. Thank you!

..., So I wondered whether anyone knew of a form with which we can calculate ΔT∗, without computing the real work ΔT. Thank you!
...
Are you asking about the change in kinetic energy in the rest frame of the center of mass? For the most general non-rigid body you have to integrate over all point-masses.

• vanhees71 and etotheipi
Are you asking about the change in kinetic energy in the rest frame of the center of mass? For the most general non-rigid body you have to integrate over all point-masses.

Yeah that's it. I wonder what such an integral would look like? Because if you use the construction $$W = \int \sum_i \tau_i d\theta_i = \Delta T$$ that gives you the total work (##\Delta T_{CM} + \Delta T*##) for general rigid body motion with torques calculated about a certain arbitrary coordinate system.

Is there any way of isolating the ##\Delta T*## in the rest frame of the centre of mass?

For the most general non-rigid body you have to integrate over all point-masses.

Actually, I've woken up now and I understand. It's just an application of the WE theorem in the centre of mass frame. Gotcha. Thanks!