A uniform circular plate of radius 2R has a circular hole of radius R cut out. Where's the center of mass?

Question

The center of the smaller hole is .8R from the center of the larger circle. [hint: use subtraction]

Answer 1

The plate (if there were no hole in it) has an area of π*(2R)^2 = 4πR^2. The hole has an area of πR^2. The center of mass is a weighted average of these two areas, with the hole treated as a negative area. Multiply the areas by their respective differences from the center of the plate, and then divide by the total area; the total area is equal to 4πR^2 - πR^2 = 3πR^2. That's [(4πR^2)*(0) - (πR^2)*(0.8R)] / (3πR^2) = -(0.8πR^3) / (3πR^2) = -0.267R. That is to say, the center of mass is at a distance of -0.267R from the center of the plate, in the opposite direction from the hole.

Answer 2

In the middle of the circle, where there is no mass.

Answer 3

okay, i have no idea. i hope this helps!