# Proof of work-energy theorem for a rigid body

Could anyone show me a way to derive the work energy theorem for a rigid body whose motion is along a fixed axis ( such as of a cylinder rolling on a plane) which states that states that $$W=\frac{1}{2} m V_{c m}^{2}+\frac{1}{2} I \omega^{2}$$ using the basic definition that $$W=\int \vec{F} \cdot d r$$.

• Related post by OP: physics.stackexchange.com/q/606476/2451 Jan 11, 2021 at 8:07
– Eli
Jan 12, 2021 at 8:51

Starting with the EOM's: \begin{align*} &\textbf{Translation}\\ &m\,\ddot{r}_{cm}=F\\ &\dot{r}_{cm}\,m\,\ddot{r}_{cm}=\dot{r}_{cm}\, F\\ &\frac{m}{2}\frac{d}{dt}\left(\dot{r}^2_{cm}\right)=\frac{d}{dt}{r}_{cm}\, F\\ &\int \frac{m}{2}{d}\left(\dot{r}^2_{cm}\right)=\int {d}{r}_{cm}\, F\\ &\frac{m}{2}\,\left(\dot{r}^2_{cm}\right)=\int {d}{r}_{cm}\, F\\\\ &\textbf{Rotation}\\ &I\,\ddot{\varphi}_{cm}=F\,R\\\\ &\text{analog}\\ &\frac{I}{2}\,\left(\dot{\varphi}\right)^2=\int {d}{\varphi}\, F\,R\\\\ &\text{thus work for the total energy ~T=\frac{m}{2}\,\left(\dot{r}^2_{cm}\right)+\frac{I}{2}\,\left(\dot{\varphi}\right)^2~is }\\ &\int {d}{r}_{cm}\, F+\int {d}{\varphi}\, F\,R= \int F\,\left(dr_{cm}+R\,d\varphi\right)=\int F\,dr\\ &\text{with}\\ &dr=dr_{cm}+R\,d\varphi \end{align*}

Remark:

F is the constraint force between the cylinder and the plane and $$~\dot\varphi=\omega$$

• Sorry dear Eli for my delayed response, I was caught up in some issues. Thank you for your response. I was however looking for a derivation that uses only the integral of $F.ds$ that would begin with it . You broke the integral from the beginning into two integrals which is not what I was looking for. Jan 13, 2021 at 14:19
• no problem, thank for your response, the last equation you see only the integral that you are looking for?
– Eli
Jan 13, 2021 at 14:22
• Yes , can you explain it Jan 13, 2021 at 15:51
• you don't have only translation motion , you have translation plus rotation motion thus $~dr\mapsto dr_{cm}+R\,d\varphi~$ see also physics.stackexchange.com/questions/606476/…
– Eli
Jan 13, 2021 at 16:51