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Body in space

This object is in space with it's centre of mass at point O. I can't understand how the object will react and rotate due to the force couple. In my mind i picture the object rotating around the midpoint of the two forces but can't prove it. How can i calculate the centre of rotation, how does the centre of mass react and why? ( Please keep the answer with minimal mathematics if possible )

Thanks in advance.

Ksekolias
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Net force is zero so no translational acceleration of the centre of mass.
The two forces which are acting are a couple which produces a torque independent of any reference point/axis and so there is only an angular acceleration about the centre of mass.

You may find the answers to this question, What is the proof that a force applied on a rigid body will cause it to rotate around its center of mass?, helpful?

The force diagram for this situation is shown below.

enter image description here

Farcher
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    Maybe I'm wrong, but it seems to me the axis of rotation of the object will be between the applied forces, and the center of mass will rotate (orbit) about the axis of rotation. – Bob D Mar 12 '20 at 11:09
  • @BobD Where is the force which causes the centripetal acceleration of the centre of mass? – Farcher Mar 12 '20 at 11:31
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    Good point, addition of diagram helped – Bob D Mar 12 '20 at 11:47
  • @BobD - You are being tricked by your intuition and thinking the center of rotation is between the forces because in real life it is almost impossible to apply a pure torque (force couple) without also constraining the body to rotate about a fixed axis. Instead of imagining forces being applied, imagine a fixed magnet subject to a torque from a magnetic field (like a free floating compass needle). It will rotate about its center of mass. – John Alexiou Mar 12 '20 at 13:45
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    @ja72 Could be. I do think visually. But when I do a little experiment on a pencil on my desk and attempt to apply a couple with the index fingers of my hands, I get a different rotation of the pencil if I apply the couple near the end of the pencil then when I apply it at the middle. In either case I get no translation, but location of the axis of rotation of the pencil changes. I know it doesn't make sense based on the physics, but there it is. Perhaps there is an alternative explanation for my observation. – Bob D Mar 12 '20 at 13:58
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    @Bob D This is exactly the reason i created this post. I kept imagining a wooden pole or a pencil being subjected to this couple. I understand that by translating these forces as forces and torque on the centre of mass i get pure torque on the centre of mass but i can't understand why i should do that. Why do i translate them to the centre of mass and not any other point on the object? ( If any other point was the centre of rotation then it would work with your observation on the pencil). – Ksekolias Mar 12 '20 at 15:17
  • Before making this i didn't know what terms to use to find good answers. I searched again and this one made me see it a bit better: https://physics.stackexchange.com/a/311616/256394 Thanks for your help. – Ksekolias Mar 12 '20 at 15:29
  • @Ksekolias I have posted an answer. It supports Farcher's answer but gives a different perspective. – Bob D Mar 12 '20 at 15:52
  • @BobD - you proved my point because by twirling a pencil you are constraining the motion with your fingers. The pencil isn't free anymore. Just look at this video to understand how looks can be deceiving. – John Alexiou Mar 12 '20 at 17:55
  • @ja72 Hmm. Are you agreeing or disagreeing with my answer? – Bob D Mar 12 '20 at 18:40
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    @ja72 The video raises a point that may be the source of my thinking that the center of rotation would be between the two forces. In the video, the torque is momentary applied by the thrower after which no external forces other than gravity act on the tennis racket. I was thinking of the couple continuing to be applied to the object, not momentarily, in which case the rotation would be between the forces. Perhaps that's what you mean by my fingers "constraining" the motion of the pencil? – Bob D Mar 12 '20 at 19:17
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I now agree with @Farcher answer, though it was (and still is) counter-intuitive to me. I am more conversant in statics and mechanics of materials then dynamics. In statics you can move a couple and it has no effect on the requirements for static equilibrium. But in mechanics of materials, where you evaluate the magnitude and location of shear and bending stresses, you cannot move the couple. That appears not to apply in the case of dynamics of rigid bodies.

You might consider the following as a way of proving that you can move the couple anywhere on the object without changing the moment about any point on the object:

What is the sum of the moments about the point exactly in between the two forces? It is 5 x 1500 = 7500 Nm clockwise.

Now take the sum of the moments about the center of mass. It is also 5 x 1500 = 7500 Nm clockwise.

In fact, if you take the sum of the moments about ANY point on the object, you will always get 7500 Nm clockwise. This shows to me that, for dynamics, the couple can be moved anywhere.

Hope this helps.

Bob D
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  • "But in mechanics of materials, where you evaluate the magnitude and location of shear and bending stresses, you cannot move the couple."-are you talking about cases when the object is not rigid? – tryingtobeastoic Mar 26 '22 at 05:31
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    Yes. Mechanics of materials deals with deformable solids – Bob D Mar 26 '22 at 07:01