Understanding Dumbbell Rotation: Laws and Material Constants

[Antti]

I just thought about a dumbbell (for some reason) and how one end rotates if I turn the other. If it was just a "mathematical system" with two flat cylinders and a long cylinder in between, then one end would rotate in exactly the same way as the other. But if the middle bar was rubber for example, then the rotation would be delayed in the other end. Now, is there some law describing the motion $$\theta(t)$$ of the other end if I know $$\tau(t)$$ (torque as function of time) of the first end. That is, if I also know the length of the middle bar and what it is made of? What material constants come into play? Could I model the bar with a simple spring instead?

 

[tiny-tim]

Hi Antti!

(nobody else has replied, so: )

Does this help … http://en.wikipedia.org/wiki/Torsion_spring?

 

[Antti]

Hi Antti!

(nobody else has replied, so: )

Does this help … http://en.wikipedia.org/wiki/Torsion_spring?

Well no. Say I had a dumbbell made entirely of Copper and I knew the exact shape of the whole thing. Then I'm thinking there should be some torsion-spring effect, though small, in the middle bar when I apply a torque to one end. No materials are perfectly rigid. How do you calculate this from the dimensions of the object and some (which?) material constants?

 

[Mapes]

Try http://en.wikipedia.org/wiki/Torsion_(mechanics)" , which gives the spring constant ($L/JG$) of a beam under torsion.