Contradiction in work-energy theorem for rigid bodies?

Question

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

Answer 1

If the force is applied at some $R$ above the CM, then $\mathrm{d}r=\mathrm{d}r_{\text{CM}}+R\omega \mathrm{d}t$, so you have an extra term. This extra term becomes:

$$ W_{\text{rot}}=\int FR\omega \mathrm{d}t=\int FR\omega \mathrm{d}\omega /\alpha=\tfrac{1}{2}I\omega^2$$

(where we replaced $\alpha=FR/I$ before integrating)

Answer 2

If the line of action for the applied force does not pass through the center of mass, then the point of application of the force experiences a tangential acceleration. This means that for a given (short) time, the dR for the force is not the same as the dR for the center of mass. This difference is not considered in the given derivation. It is not valid if rotation occurs. Basically, the extra distance traveled by the force does the work to supply the rotational energy.