Is it possible to vectorially add and find the resultant of several forces acting on different points of an extended body? For example, if I apply a couple (equal and opposite forces) to the two ends of a rigid rod, can I claim that the net force acting on the rod is zero? All I know form vector addition that vectors can be added which act at the same point/particle.
EDIT: I think, we cannot add forces on a rigid body acting at different points. If we could, then the net force on the rod would be zero, and assuming Newton's 2nd law is applicable, then it says the acceleration of the rod it zero. But the rod rotates and therefore, various points on the rod do accelerate. One might say that the centre-of-mass doesn't accelerate. However, if we apply Newton's second law directly to a rod, one has no clue whether the acceleration $\textbf{a}$ sitting on the RHS of $\textbf{F}=m\textbf{a}$ is really that of the centre-of-mass. And to arrive at the concept of centre-of-mass one really have to use Newton's third law independent of Newton's second law (as it is done for a system of N-particles).
Best Answer
Newton's second law, when applied to point particles, states that there will be no motion if the sum of applied forces equals zero. Since a rigid body is composed of infinitely many points particles, there will be no motion if and only if the sum of applied forces on each and every point particle the body is composed of is zero.
In the context of your question, saying the resultant force on your rod is zero would be pretty misleading. Mathematically, if you sum those vectors the sum will be zero, but you are considering that you can move your vectors freely and nothing changes. Since you now have a set of particles, each of them acts differently, so you cannot detach those force vectors from its associated points. Indeed, if you apply equal and opposite forces to opposite ends of a rod, even though summing them would give you zero, you will create a torque that will rotate the rigid body around its centre of mass. The total force applied to the system is zero, but this doesn't mean internal forces play no role, because the system is composed of smaller systems that are not isolated (namely, the particles).