- #1

etotheipi

*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!