# Massive Pulley Dilemma

1. Apr 22, 2014

### dkb12085

Hi all:

I am studying the mechanics of a system where there is string wrapped around a massive pulley, and a block at the end of the string.

Correct me if any of this is wrong, but the system (pulley and block) must have the same linear acceleration. So, in the case that the pulley is much more massive than the block, it will require greater torque, and subsequently force to give it the same acceleration.

How is this accounted for? Does the string have different tension around the pulley than it does at the block end?

Isn't it true that Tension= mg-ma because it is exerting upward force on the block?

Thank you for your efforts in explaining this!

Thank you,
dkb12085

2. Apr 22, 2014

### 256bits

If you consider the string to be massless then the tension at the pulley end will be the same as at the block end.

For real strings, if short or lightweight, you can make the assumption that the string mass is negligable in comparison the the block.

If the string length is longer or more massive, then the mass of the string and the mass of the block both contribute in the acceleration of the pulley. As one moves up the string, starting from the block, the mass below a particular position increases due to the addition of the incremental mass of the string below. For that reason, the tension as one moves up the string, where the mass of the string is not negligable, also increases.

Like you said, the block, the string, and the contact of string at the pulley all have the same linear acceleration.
While the block has translational inertia, as evident by the formula F=ma ( a more massive object requires more of a force for the same acceleration as a less massive object ), the pulley has a corresponding rotational inertia designated as moment of inertia ( mass moment of inertia ) about the axis of rotation.
http://en.wikipedia.org/wiki/List_of_moments_of_inertia

The corresponding rotational equation is T = I $\alpha$, ( $\alpha$ is Greek alpha symbol lower case )
where T = torque, I = mass moment of inertia, a = angular acceleration
http://theory.uwinnipeg.ca/physics/rot/node5.html