How is the work done by the internal forces acting in a rigid body zero?
Actually I read in a book an example for the same.
Let me present that example here.
Consider a rigid body having two particles $A$ and $B$. Suppose, the particles move in such a way that the line $AB$ translates parallel to itself. The displacement $d\textbf{r}_A$ of the particle $A$ is equal to the displacement $d\textbf{r}_B$ of the particle $B$ in any short interval of time. The net work done by the internal forces ,i.e, the force that $A$ exerts on $B$ and the force that $B$ exerts on $A$, is zero.
How can it be analysed mathematically that the work done is zero?
This example is very unclear to me.
Best Answer
Your confusion might be coming from not clearly understanding that you need to define a system, and then all of your quantities are referenced to the system you have defined.
If your system is A and B, then the force that A applies to B is by definition an internal force, and no work is done by that force on the system. But you can choose your system however you would like. Nature doesn't care where you draw your boundary. If you take the system to be A, the the force of B on A is external, and work can be done on the system. Note that it's not the same system as the first case!
What's internal and what's external is a matter of bookkeeping. The setup of the equations will be different, but the final answer will always be the same.