- When pushing down on a lever to lift something heavy, your push is a force. You apply it at some point.
Now consider the exact same situation, just with gravity being the force instead of your push:
- gravity pulls down on the lever. It applies this pull at some point.
This point is the center of gravity. Let's call it CoM. That is all. You talk about the CoM in connection to where gravity pulls.
How do moments help me find the centre of gravity?
Only moments caused by gravity can help you, since only then does the moment have anything to do with the CoM.
Gravity will pull in the CoM as a force applied at a point. So as long as this CoM is not straight under the rotation point, gravity will try to rotate the object (it will cause a torque, since the distance $r$ is not zero in your formula, unless the CoM is exactly under the rotation point).
So when your object hangs still and doesn't rotate anymore, you know that the CoM must be somewhere directly below the rotation point. On the vertical line below.
If you now hang the object in another point so that you have a new rotation point, then when the object hangs still again, you again know that the CoM must be directly below somewhere on the vertical line under the new rotation point. These two lines have only one point in common. So this point must be the CoM, because it must be the same for both situations (since you didn't redistribute the mass).
What forces are being applied to the piece of card?
If it hangs freely in a hook for example, so it can rotate freely, then only two forces act on the card. Gravity from the CoM and the hook's normal force holding it.
Since the normal force by the hook works in the rotation point, this gives no torque (the distance $r$ is zero). Therefore only gravity is left to do a torque to make it rotate.
And if the CoM is vertically below the rotation point, then also gravity gives no torque as explained above. Therefore the above method to find the CoM will only work if no more forces than the gravity can cause a torque.