Physics questions center of mass?
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!
- NCSLv 72 months agoFavorite 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
- SlowfingerLv 62 months ago
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)
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)
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
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