What creates the angular acceleration in a pulley?

[conorwood]

Looking at the free body diagram of a fixed pulley with a rope going over the top half, the diagram would show tension going downward on either side and an anchor going up equaling twice the tension. I understand this is this case so the pulley doesn't move in any rectilinear motion. However, since the tensions are equal, the torque cause by the forces of tension are equal and opposite and they cancel out. I am having trouble understanding what causes angular acceleration in this case, for example when two masses of different size are attached to the ends of the rope.

Thanks

 

[sophiecentaur]

When the rope is accelerating, the tensions are NOT equal. There is more tension in the direction the rope is accelerating. Not a lot, of course, as pulleys are deliberately light, usually.

 

[conorwood]

If that is the case, then why can we analysis the forces in something like an Atwood machine as if the tension on either side were equal?

for example, in the case where we know m1 and m2 (mass one and mass two), m1<m2, and T = force of tension

for m1:

ƩF = m1a

T - m1g = m1a

and for m2:

ƩF = m2a

m2g - T = m2a

we can solve for T in one equation and substitue this in for the other to find acceleration.

If the two T's are different like how sophiecentaur describes, then how are we able to make this substitution?

 

[Nugatory]

we can solve for T in one equation and substitue this in for the other to find acceleration.

If the two T's are different like how sophiecentaur describes, then how are we able to make this substitution?

That substitution is completely correct only if the pulley is massless and frictionless. However, it is a very good approximation if the mass of the pulley and the frictional effects are small compared with the masses of the two weights.

You could, if you wanted, reject this approximation and do a more exact solution. The tension on the two sides is not the same, it's slightly different, and this difference is caused by a small force between the rope and the edge of the pulley. It is this force that applies torque to the pulley to change its rotation. Solve the problem this way and you'll end up doing a huge amount of additional math, and all you'll have to show for it is a few extra decimal places in the predicted acceleration of the masses.

 

[conorwood]

thank you!

 

[sophiecentaur]

Solve the problem this way and you'll end up doing a huge amount of additional math, and all you'll have to show for it is a few extra decimal places in the predicted acceleration of the masses.

Actually, it's not a massive problem (no pun intended) if you treat the pulley as being equivalent to just another mass that needs to be accelerated (say by adding half the eqivalent to each side). To find that equivalent mass, you 'just' need to be familiar with Moment of Inertia.