We've a rigid body at rest. A force acts on it which leads it to translate( velocity 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