Physics questions center of mass?

Question

The location of the center of mass of the partially eaten, 12-inch diameter pizza shown in the figure is Xcm = - 1.5 in

and Ycm = -1.5 in . 

A) Assuming each quadrant of the pizza to be the same, find the center of mass of the uneaten pizza above the x axis (that is, the portion of the pizza in the second quadrant). Find the x-coordinate.

B) Find the y-coordinate.

Best answer will be given asap, thank you for your help!

Answer 1

Favourite answer

I think this can be reasoned out without much calculation. In order for the missing ¼ of the pizza to "pull" the CM of the uneaten part to (0, 0), that missing part must have a CM at

3*(1.5, 1.5) in = (4.5, 4.5) in

Then for that ¼ of the pie in QII, (x, y)_cm = (-4.5, 4.5) in

Answer 2

The center of mass of the system is given by

x = sum of moments / total mass ............ (1)

Let xc2 and yc2 are coordinates of the center of II quadrant part

Analogous, centers of III and IV quadrant parts are designated

(xc3, yc3)  and (xc4, yc4)

Let M is the mass of 1/4 pizza

The center of mass of the system is

Xcm = (M xc2 + M xc3 + M xc4) / (3M)

Xcm = (xc2 + xc3 + xc4) / 3 .......... (2)

Assuming each quadrant of the pizza to be the same, because of symmetry, centers of masses in quadrants III and IV are

(xc3, yc3) = (xc2, -yc2) and

(xc4, yc4) = (-xc2, -yc2)

substitute these values in equation (2)

Xcm = (xc2 + xc2 - xc2) / 3 = xc2 / 3xc2 = 3Xcm = 3*(-1.5) = -4.5

in a similar way, for y

Ycm = (yc2 - yc2 - yc2) / 3 = -yc2 / 3yc2 = -3Ycm = -3*(-1.5) = +4.5

The center of II quadrant part is(-4.5, 4.5)

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Alternative method:

Above, we add up 3 quarters to get 3/4 pizza. We could also subtract 1/4 from the whole.

If a missing part is added then the center of mass of the whole would be at (0,0)

in that case

Xcm = (4M * 0 - M xc1) / (3M)

where

M is the mass of 1/4 of pizza

xc1 is x-coordinate of the center of the eaten quarter

Xcm is the center of mass of the 3/4 pizza

Xcm = -xc1 / 3

xc1 = -3Xcm

xc1 = -3 * (-1.5) = 4.5 in

Because of symmetry with respect to y-axis, for the second quadrant part

xc2 = -xc1 = -4.5 in

similar 

yc1 = -3Ycm

yc1 = -3 * (-1.5) = 4.5 in

Because of symmetry with respect to the y-axis, for the second quadrant part 

yc2 = yc1 = 4.5 in