Does Pulley Mass Affect Rope Tension Symmetry?

[iva]

Hi there,

If a pulley system made of 2 same mass blocks (on a a table, one hanging), a disk pulley with mass but no friction on the string nor pulley axis, does it mean that the Tension on the rope is the same on either side of the rope? Everywhere I've read implies this, and it makes sense to me because the rope is 1 continuous object and if the pulley is frictionless then surely there is nothing that should make the tension be different on either side of it? But I'm not sure if the mass of the pulley should affect this.

And then what if the pulley was massless, surely in that situation the tension should be the same throughout?

Thank you

 

[Doc Al]

If the pulley is massless and frictionless, then the tension will be the same throughout the string (treating the string as massless). But if the pulley has mass, then a net torque is required to accelerate it--so the tension will be different on each side of it.

(When we speak of a frictionless pulley we mean no friction on its axis. There must still be static friction of string on pulley, otherwise the pulley will not turn.)

 

[iva]

Thanks, but do you still need to consider Torque even if there is no friction on the rope going over the pulley? I'm just thinking that if you have this pulley that just rolls easily (I'm thinknig of some sports equipment where often the rope doesn't go over the pulley because there is dirt on it, but when i rub sillicone over the pulley and clean it it rolls perfectly)

Also if the string doesn't slip on the pulley, doesn't this mean that Linear accel = radius * angular accel, meaning that the acceleration on both sides would have to be the same?

 

[Doc Al]

Thanks, but do you still need to consider Torque even if there is no friction on the rope going over the pulley?

If you really mean 'no friction on the rope', then the tension will be same on both sides of the pulley. But the pulley will not turn! It's as if it wasn't there. Its mass is irrelevant. This is generally not what is meant by a 'frictionless pulley'.

I'm just thinking that if you have this pulley that just rolls easily (I'm thinknig of some sports equipment where often the rope doesn't go over the pulley because there is dirt on it, but when i rub sillicone over the pulley and clean it it rolls perfectly)

What makes a pulley roll easily is no friction on its axis, not no friction between the rope and the pulley.

 

[iva]

Thanks, OK i realize I'm mixing up and confusing friction with slipping ( as I've just learned about the slipping factor that if there is no slipping then linear acceleration is related to angular acceleration by radius * angular acceleration)
So if there is no slipping of the rope on the pulley doesn't the formula

linear acceleration = radius * angular acceleration

then suggest that the acceleration on either side of the pulley will be the same? and if the mass of the 2 blocks are the same then that should give the same tension?

Thanks again :)

 

[Doc Al]

Thanks, OK i realize I'm mixing up and confusing friction with slipping ( as I've just learned about the slipping factor that if there is no slipping then linear acceleration is related to angular acceleration by radius * angular acceleration)
So if there is no slipping of the rope on the pulley doesn't the formula

linear acceleration = radius * angular acceleration

then suggest that the acceleration on either side of the pulley will be the same?

Even with slipping the acceleration of the rope (and attached masses) would be the same (in magnitude).

and if the mass of the 2 blocks are the same then that should give the same tension?

No. Assuming the pulley has mass, then the tension on each side of the pulley cannot be the same.

 

[iva]

So if the acceleration is the same either side, then what this means is that the Tension is a result of the pulley mass PLUS each of the block masses separately?? Sort of like

T1 = (Mass block1 + mass pulley) * linear acceleration and

T2 = (Mass block2 + mass pulley) * linear acceleration ?

This result doesn't make sense though, because if the masses are the same I'm going to end up with the 2 tensions being the same and with the one block hanging from the pulley it can't be

thanks for your patience

 

[Doc Al]

So if the acceleration is the same either side, then what this means is that the Tension is a result of the pulley mass PLUS each of the block masses separately?? Sort of like

T1 = (Mass block1 + mass pulley) * linear acceleration and

T2 = (Mass block2 + mass pulley) * linear acceleration ?

This result doesn't make sense though, because if the masses are the same I'm going to end up with the 2 tensions being the same and with the one block hanging from the pulley it can't be

Not sure where you got those equations. The angular acceleration of the pulley is governed by:
Net Torque = I*alpha

The net torque will be: (T1 - T2)r

Only the two masses are linearly accelerated. You need three equations--one for each object--to describe things. Then you can solve for the acceleration and the tensions.

 

[iva]

Thanks yes i got the rotational equation when i started working this out
but then got confused with the tensions , wanting to make the tension for both sides the same as well as the accelerations, but now it makes sense. The pulley system is not as intuitive or straight forward as I thought it would be

Thanks for helping me with this