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If you have some random object at rest and you apply a couple to it, the net force acting on it is zero. However because a moment acts on it, it starts to rotate.

So you had an object at rest, a net force of zero was applied to it, and it is now moving. Why doesn't this violate Newton's First Law?

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    $\begingroup$ You're still applying a net torque, the rotational analog of a force. Therefore, the object will change its angular momentum (start to rotate). $\endgroup$
    – Danu
    Oct 12, 2013 at 20:31
  • $\begingroup$ @Danu But Newton's Law says the net force has to be non-zero, not torque. The torque isn't zero, but the force is. Isn't that a violation? $\endgroup$
    – dfg
    Oct 12, 2013 at 20:33
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    $\begingroup$ The forces don't both act on the same (part of the) thing, and therefore Newton's laws don't tell us they should cancel out. $\endgroup$
    – Danu
    Oct 12, 2013 at 20:41
  • $\begingroup$ @Danu Buts its a rigid body, so can't the whole thing be treated as one "thing"? $\endgroup$
    – dfg
    Oct 12, 2013 at 20:47
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    $\begingroup$ You can definitely treat the rigid body as a system, but then the only thing that you can conclude from the fact that the net force on the system is zero is that the center of mass won't move. See my answer. $\endgroup$ Oct 12, 2013 at 20:56

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Newton's first law of motion for a point particle states that a particle at rest will stay at rest and a particle in motion will stay in motion unless acted on by an unbalanced force. In other words, if the net force on the particle is zero, then the velocity of the particle will stay constant. Newton's first law of motion for a system of particles states that if the net external force on a system is zero, then the velocity of the center of mass of the system will remain constant. It says nothing about the velocity of each of the particles.

So if the center of mass of the rigid body is initially at rest, and there's no net external force, then the center of mass will continue to be at rest. But that doesn't mean that the individual parts of the rigid body will remain at rest. There's two ways to think about this. One way is to apply Newton's first law to each part of the rigid body: if two forces act on the rigid body, but they act on two different places, then one part of the rigid body will only experience one of the forces, so it can move. The other way to think about this is to use the angular version of Newton's first law: if the net external torque on a rigid body is zero, then the angular velocity of the rigid body is constant. Since there's an external torque in our example, there's no requirement that the angular velocity must be constant, so the rigid body can rotate even though it wasn't rotating before.

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