hi Maiya

post #24......

in what method is the centroid found there........

This comes from a physics topic called moments.

I'll show with a simple example first.

Suppose we have a 10 cm rod, attached to a pin at A, free to rotate about A, and with weights attached at various distances from A. (see diagram)

Each weight will have a turning effect. The further a weight is from A the more turning effect it will have. For example, the 2 Kg weight that is at B will exsert more turning effect than the 2 Kg weight that is only 3 cm from A

The 'moment' of a weight's turning effect is found from this formula:

and the total moment is found by adding these up.

Now suppose I want to replace the four weights with a single weight of 8 Kg (the total weight). Where should I hang it so that it has the same turning effect?

That is where the centroid of those weights is. It is the point where, from a mechanical point of view, you may consider all the weight to lie.

In calculations involving Newtonian mechanics, you may consider the weight of 8 Kg to be at that distance in all subsequent calculations.

For example, if I want to push up at B with sufficient force to hold the rod in a horizontal position then the force (for simplicity I'll stick to Kg for this) would be

I am making a couple of assumptions here so I'll say what they are:

(i) I'm using weights in Kg whereas a true weight should be measured in Newtons. But as the conversion factor (9.81) would cancel out throughout the problem, I can leave this out to keep things simple. In your centroid problem, I went one step further and used areas rather than weights, as I assumed the boomerang would have constant density across the surface.

(ii) I haven't allowed for the rod having any weight. We can say it is a 'weightless' rod. Of course that cannot be true in reality, but trying to allow for the centroid of the rod as well would have just introduced an unnecessary extra calculation and made the process harder to follow.

For the boomerang, the weight is spread out across a continuous surface. So it is necessary to use integration to summ up all the little bits of area and also to sum up all the moments of the little bits. Then you can use the formula

Bob