We've a rigid body at rest. A force acts on it which leads it to translate ( velocityvelocity of center of mass is $V_{cm}$ )and rotate about a fixed axis with $\omega$
Kleppner and Kolnekow prove that:
Work done by the force equals the change in kinetic energy of the center of mass. So $W=\Delta K_{cm}=\frac{1}{2} m V_{c m}^{2}$ and here is their proof
To derive the translational part, we start with the equation of motion for the center of mass $$ \begin{aligned} \mathbf{F} &=M \frac{d^{2} \mathbf{R}}{d t^{2}} \\ &=M \frac{d \mathbf{V}}{d t} \end{aligned} $$ The work done when the center of mass is displaced by $d \mathbf{R}=\mathbf{V} d t$ is $$ \begin{aligned} \mathbf{F} \cdot d \mathbf{R} &=M \frac{d \mathbf{V}}{d t} \cdot \mathbf{V} d t \\ &=d\left(\frac{1}{2} M V^{2}\right) \end{aligned} $$ Integrating, we obtain $$ \oint_{\mathbb{R}}^{R_{i}} \mathbf{F} \cdot d \mathbf{R}=\frac{1}{2} M V_{b}^{2}-\frac{1}{2} M V_{a}^{2} $$
But we also know from work kinetic energy theorem that the Work done by an external force equals the change in kinetic energy of the body so $W=\Delta K=\frac{1}{2} m V_{c m}^{2}+ \frac{1}{2} I \omega^{2}$.
Then these two equations contradict each other!
Can anyone please help me. I'm not able to sleep
