Understanding the Mathematics Behind the Center of Gravity Problem

[unctarheels1]

I am trying to understand the math for this problem. My instructor did a poor job of explaining it in class.

He took a "skyhook" or belt hanger ( looks like a music note make out of wood)" put a stiff belt on it and balanced it on the edge of a table. He exaplained that the center of gravity of the belt (which was at an angle and leaning under the table) was directly under the base of the belt hanger and that made it balance.

There has to be a mathematical answer to this. Can someone help?

 

[dx]

The center of mass is a kind of average position of the mass. In many cases, it is useful to consider the body to be concentrated at the center of mass. Mathematically, the position of the center of mass is

$$\frac{\sum{m_{i}}{r_{i}}}{M}$$

$$m_{i}$$ is the mass of the ith particle and $$r_{i}$$ is its corresponding position.

 

[unctarheels1]

here is a link to a picture of the problem I am describing.

http://www.uvm.edu/~dahammon/demonstrations/balancingbelt.html

 

[unctarheels1]

here is a link to a picture of the problem I am describing.

http://www.uvm.edu/~dahammon/demonstrations/balancingbelt.html


If I could get a correct Free body diagram for it, I think I would understand it.

- You should have a force acting upward from the table to the tip of the "belt hanger" and then the belt puts a force on the hanger as well. I suppose it would act at the angle the belt is hanging. Am I missing anything else other than the gravity, which would act on the entire wooden piece?

 

[dx]

When you balance something at its center of mass or directly under or over its center of mass, there is no torque. Thats why it doesn't turn. As for the mathematics of it, let's see what information we can get from the situation. We need an expression for the torque. Let's take the horizontal axis as the x axis. the torque would be

$$\tau = \sum{m_{i}}g{x_{i}} = g\sum{m_{i}}{x_{i}}$$

Where the $$x_{i}$$s are the distances from the fulcrum.
But $$\sum{m_{i}}{x_{i}}$$ is the total mass $$M$$ times the position of the center of mass. Since in our case, the position of the center of mass is 0, i.e it is at a distance $$x = 0$$ from the fulcrum, the torque must be zero.