Answer to Question #201677 in Physics for minahil

Question

Figure shows a uniform metal plate of radius 2R from which a disk of radius R has been stamped out (removed). a) Using the x-y coordinate system shown, locate the center of mass of the plate. b) Now the removed disk is welded to the plate as shown in the figure. Find the center of mass of the combined system

Answer 1

a) We know that the center of gravity of the smaller disc at its center, R away from the center of the whole disc.

Assume that the center of gravity of the large disc with the circle cut off is x away from the center of the whole disc, so, we can write

$$x_x=\frac{x_{\text{small}}A_{\text{small}}+x_{\text{with hole}}A_\text{with hole}}{A_\text{total}}.\\\space\\ 0=\frac{R\pi R^2-x[(\pi (2R)^2-\pi R^2]}{\pi(2R)^2},\\\space\\ 0=\frac{R^3-3R^2x}{4R^2},\\\space\\ x=\frac R3.$$

In our coordinate system, it is -R/3.

b) Draw the figure:

Find the center of mass:

$$x_c=\frac{[\pi( 2R)^2-\pi R^2](-x)+\pi R^2(-3R)}{\pi (2R)^2},\\\space\\ x_c=\frac{-3x-3R}{4}=-R.$$