Equilibrium of a Rigid Body question

[Lamoid]

The solid uniform disk of radius b shown can turn freely on an axle through its center. A hole of diameter D is drilled through the disk; its center is a distance r from the axle. The weight of the material drilled out is Fwh. Find the weight of an object hung from a string wound on the disk that will hold the disk at equilibrium in the position shown.

I really can't figure this one out. I use the center as the axle but I can't find what the distance from the axle would be for the center of mass since the hole drilled in moves it from the center. Can anyone give me a hint to get me started?

 

[Doc Al]

Hint: A hole (no mass) can mathematically be thought of as the sum of a positive mass plus a negative mass.

 

[Lamoid]

So I want to set up an equation where the torque from the weight is equal to the torque of the full disk minus the torque of the hole?

 

[Doc Al]

The net torque must be zero. The torque from the "negative mass" hole will be in the opposite direction than that of a positive mass "hole".

 

[Lamoid]

OK so i work out the equation

(Fw)= - (r/b)(Fwh) sin theta.

Unfortunately, the answer has it being cos theta as well as there being no negative sign which makes more sense.

How I worked it out was

(Fw)(b) sin 90 = torque of the plate with hole = torque of the entire thing - torque of hole

(Fw)(b) = 0 - (Fwh)(r)(sin theta)

(Fw)= - (r/b)(Fwh) sin theta

Where did I go wrong?

 

[Doc Al]

How I worked it out was

(Fw)(b) sin 90 = torque of the plate with hole = torque of the entire thing - torque of hole

The net torque must be zero, so that should be:
(Fw)(b) sin 90 + torque of the plate with hole = 0
or:
(Fw)(b) sin 90 + torque of the entire thing - torque of hole = 0

That will get rid of the negative sign.

(Fw)(b) = 0 - (Fwh)(r)(sin theta)

(Fw)= - (r/b)(Fwh) sin theta

Why are you using sin(theta)?

 

[Lamoid]

Should it be sin (90 - theta)?

Edit: I mean sin (90 + theta) which would be equal to cos theta correct?

 

[Doc Al]

Should it be sin (90 - theta)?

Edit: I mean sin (90 + theta) which would be equal to cos theta correct?

I'd say the first, which does equal cos(theta).