The lower boundary of the plate is defined by: y =0.750 x 2 . Find the distance
ID: 2148554 • Letter: T
Question
The lower boundary of the plate is defined by:y=0.750x2. Find the distance from the rounded tip of the plate to the center of mass.
Calculate the x-coordinate of the center of mass of the remaining pizza. The grid units are inches. Note that the tip and corners of the slice which was removed lie exactly on intersection points. Your answer must be accurate to within 0.01 in (inches) or 0.0254 cm.
Explanation / Answer
x(c.m.) = (1/M)integral x*dm M = mass of pie after removal of wedge. For pie before the wedge is removed, x(c.m.) = 0 After the wedge is removed, x(c.m.) = 0 + (1/m) *integral of x*dm from x = -6 to x = 0 and bounded by y1 = x/6 and y2 = -x/6 where m = mass of wedge (This equation is valid because the pie is of uniform surface density). Now, dm = s(y1 - y2)dx = (sx/3)dx So x(cm) = (s/m)*integral of [(x^2)/3]dx from x = -6 to x = 0 = (s/m)*24 But s/m = 1/area of wedge = 1/[(R^2/2)(2pi - 2d)] = 1/[(R^2)(pi - d)] where 2d = wedge angle = 2 arc sin (1/6) = 0.3349 rad or d = 0.16745 rad So x(cm) = 24/(R^2*(pi - d) = 0.224" answer
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