# Couple force in a rotating steel ruler

I learnt that pure rotational force can be caused by only couple force , two forces acting in opposite direction which are equal in magnitude in 2 different points at a system.

But I am trying to understand how a couple force is created in this steel ruler as shown in this video

The scale is bent in a way such that only the center region is in contact with the table. Then If I give force to one side of the ruler , it rotates about the center region.

How is the opposite force produced in the ruler so that it rotates about the axis ? My Intuition says its friction due to table because no other factor is affecting the rotation but I'm finding it difficult to pin point the exact process

• Friction opposes the applied force to keep the center of mass from moving. Apr 23 at 21:12

Since the center of mass did not translate when the impulse was applied to the end, from Newton's 2nd law the static friction force during the applied impulse must be equal in magnitude and opposite in direction to the impulse force. By definition, this is a couple (a.k.a. force couple, moment of a couple).

A couple is called a "free vector", meaning it can be moved anywhere on the body and have the same external effect on the body. A couple consisting of two equal and opposite forces $$F$$ separated by a distance $$d$$ has the same external effect (pure rotation) as a couple of two equal and opposite forces $$F/2$$ separated by a distance $$2d$$. Each couple has a value of $$Fd$$

Hope this helps.

The impulse you imparted with your finger at the end of the ruler happens to be smaller than or equal to the maximum impulse at short times that the frictional interaction at the part of the ruler in contact with the table can produce. Then the combination of both impulses, separated at that distance, is the couple that gives the angular momentum to the ruler.

The use of the couple is much more conceptual and widely applicable than that example.

• The impules created are acting in center and end of the ruler , however doesn't couple need to act on both ends so that we get a rotation ? Apr 23 at 18:25
• No need. Pure rotation as long as you can get a couple. Apr 23 at 18:28
• Then the impulse actually carries towards the end of the ruler to induce a force so that particles in that position rotate ? Apr 23 at 18:31
• Oh, that is definitely the case. But we would never use that kind of language. Still, shearing force, or transfer of momentum, is somehow a point of contention for people. Apr 24 at 2:20

The point at which the ruler touches the table acts as a pivot with friction not allowing translational motion.

When a force is applied to one end of the ruler another (frictional) force equal in magnitude but opposite in direction acts at the pivot.
There is your couple acting on the ruler.