There has to be a force on the chain. If there was no force, the momentum of the links that are going straight would keep them going straight. But as you illustrated, they go around the sprockets. The centripetal acceleration of the links makes chain want to come off the sprocket as it goes around. The chain, being a loop, prevents this because the other side of the chain going around the other sprocket wants to come off in the opposite direction. All this results in tension in the chain.
If the chain was a perfect circle, then there would be no force on it.
If the chain was a perfect circle, then there would be no force on it.
Relative to what?
Is the chain moving?
Imagine an observer travelling around a stationary sprocket / chain system. To the observer the chain would appear to be moving and there would be forces on the chain relative to the forces on the observers vehicle that would be zero relative to the chain.....
Here we go....
Is the chain moving?
Imagine an observer travelling around a stationary sprocket / chain system. To the observer the chain would appear to be moving and there would be forces on the chain relative to the forces on the observers vehicle that would be zero relative to the chain.....
Here we go....
Guys,
Instead of a chain, picture a rail road track of the same shape. Fill the track ALMOST entirely with cars, frictionless of course, in an airless world, with an inch or two between them.
Now set all the cars in motion... exactly the same speed... yada... yada... yada. Any tension in chain?
When you say "tension in the chain"... where exactly in the chain are you talking about?
There is no net work being done... that doesn't mean there isn't tension somewhere.
Instead of a chain, picture a rail road track of the same shape. Fill the track ALMOST entirely with cars, frictionless of course, in an airless world, with an inch or two between them.
Now set all the cars in motion... exactly the same speed... yada... yada... yada. Any tension in chain?
When you say "tension in the chain"... where exactly in the chain are you talking about?
There is no net work being done... that doesn't mean there isn't tension somewhere.
If you replace the chain with a railroad track and put cars on it, and then set the cars in motion (I'm assuming the cars are following the track), then the cars will fly off the track as the track starts to curve around the sprocket. You never said to set the track in motion too, so I assumed that was stationary. 
Now let's say that you mount the cars to the track, and set the whole thing in motion, then the cars will want to pull the track (chain) off the sprockets due to centripetal acceleration. The track (chain) is designed to not come apart, therefore, the cars will produce tension in the track (chain). The straight parts of the chain, connecting both sprockets, will have tension in them too, because the two parts of the chain, around the sprockets, are trying to separate from each other.
Now let's say that you mount the cars to the track, and set the whole thing in motion, then the cars will want to pull the track (chain) off the sprockets due to centripetal acceleration. The track (chain) is designed to not come apart, therefore, the cars will produce tension in the track (chain). The straight parts of the chain, connecting both sprockets, will have tension in them too, because the two parts of the chain, around the sprockets, are trying to separate from each other.
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