Can't get this tension derivation........

[singh_abhi]

Ok guys, i just don't seem to know anything about tension, i thought that it would always equal the weight that is suspended from the rope...But that does not seem to be the case...and also i don't get the issue of its direction.
Anyways, suppose we have an atwood machine and two unequal masses attached to a rope...then the system accelerates...What I don't get is the fact that here Tension is not equal to any of the weight, why is it not so...When we suspend a single weight from a ceiling, then it is...i think the reason i am confused is that i have no understanding of the fundamentals of tension...Any help will be appreciated. Thanks...

 

[berkeman]

Ok guys, i just don't seem to know anything about tension, i thought that it would always equal the weight that is suspended from the rope...But that does not seem to be the case...and also i don't get the issue of its direction.
Anyways, suppose we have an atwood machine and two unequal masses attached to a rope...then the system accelerates...What I don't get is the fact that here Tension is not equal to any of the weight, why is it not so...When we suspend a single weight from a ceiling, then it is...i think the reason i am confused is that i have no understanding of the fundamentals of tension...Any help will be appreciated. Thanks...

Are you familiar with drawing free body diagrams (FBDs)? They will show all of the forces on an object. If the rope is not just supporting the weight but is also providing acceleration upward, the tension force will be greater than the weight being accelerated...

 

[singh_abhi]

Are you familiar with drawing free body diagrams (FBDs)? They will show all of the forces on an object. If the rope is not just supporting the weight but is also providing acceleration upward, the tension force will be greater than the weight being
accelerated...

I understand it, but only from a mathematical point of view, since there is an upward acceleration on the smaller block, and since weight acts downwards ,then the upward acceleration must be due to tension. I also know that the net force acting on the system would be m2g-m1g...and that this would equal (m2+m1)a...
I get why this happens conceptually...I just can't get why tension is less than the heavier weight and greater than the smaller weight.

 

[CWatters]

I just can't get why tension is less than the heavier weight and greater than the smaller weight.

Look at the forces on each mass on it's own...

The heavier weight M₁ is accelerating downwards suggesting that the tension in the rope isn't sufficient to support it's weight... T₁ = M₁(g-a)

The lighter weight M₂ is accelerating upwards so the tension must provide an upward force greater than it's weight... T₂ = M₂(g+a)

If the same rope supports both masses (and any pulley is frictionless with negligible inertia) then the tension is the same in all parts of the rope so equate..

T₁=T₂
and
M₁(g-a) = M₂(g+a)
expand
M₁g-M₁a = M₂g+M₂a
rearrange
M₁g-M₂g = M₁a+M₂a
or ...
a = (M₁-M₂)g/(M₁+M₂)

Note that a is less than g which is what you would expect. The small mass stops the larger one falling at g.