"How Fr=Ialpha Works in a Pulley w/ Mass

[JiggaMan]

Homework Statement

How does Fr=Ialpha work and how is it applied to a pulley with one mass attatched?

Homework Equations

F * r = I * alpha
I = moment of inertia

The Attempt at a Solution

I'm assuming F would be the tension of the pulley with the string. But how does that equation work? can someone explain the mechanics of it?

 

[Dopplershift]

F is the tangetial forice
r is the radius (in this case the radius of the pulley)
I is the moment of inertia (assume the pulley is a soild disk, so I = 1/2mr^2
alpha is the angular acceleration.

 

[Cutter Ketch]

That equation is accounting for the fact that the pulley itself has inertia. Let's say you have a mass on a rope over a frictionless pulley (the bearing is frictionless, not the rope groove!) and you are holding one end of the rope. Nothing is moving, and nothing is accelerating, so it doesn't matter that the pulley has inertia. Now suppose you let go of the rope. Gravity pulling down on the mass starts to accelerate the mass. However it ALSO starts to accelerate the pulley. As long as the rope doesn't slip, the pulley speed has to match the acceleration of the bucket. The applied force is m g. If the acceleration of the mass is a, then the angular acceleration of the pulley must be α= a/r. That means that:

M g = M a + α I
=> Mg = Ma + a I / r
=> a = g / (1+ I / (M r))
Taking the pulley to be a solid disk
=> a = g / (1 + (m/(2M)) r)

So the larger the radius or the higher the mass the more the inertia of the pulley slows down the free falling mass. This is one example of how the inertia of the pulley is used. I hope that helps.

 

[haruspex]

I'm assuming F would be the tension of the pulley with the string.

In case it is not clear, in general, F would be the difference between the tensions on each side of the pulley.

 

[CWatters]

I'm assuming F would be the tension of the pulley with the string. But how does that equation work? can someone explain the mechanics of it?

F * r = I * alpha

F * r = Torque

so it's saying..
Torque = Moment of Inertia * Angular acceleration

Compare that to Newtons law for the linear case...
Force = Mass * Acceleration

Note that it's the Net Force or Net Torque that matters in the above. The net torque due to a belt on a pulley is equal to the difference in tension on each side of the pulley as Haruspex said.