Question

Learning Goal: To be able to find the center of gravity, the center of mass, and the centroid of a composite body. A centroid is an object's geometric center. For an object of uniform composition, its centroid is also its center of mass. Often the centroid of a complex composite body is found by, first, cutting the body into regular shaped segments, and then by calculating the weighted average of the segments' centroids. An object is made from a uniform piece of sheet metal. The object has dimensions of a = 1.40 ft , where a is the diameter of the semi- circle, b = 3.69 ft , and c = 2.35 ft. A hole with diameter d=0.750 ft is centered at (1.14, 0.700).

Answer 1

0.750 ft. Centre of hole (114,0.7002 2.35fé o Toft * TN- ott 3.69 ft we know Bc² = De² - BD² = 2.352 -1.402 B = 3.562 - BC = 1.89 ft I A B = 3.69 -0.70-1.89 = 1.Ift A4 = 4x 1.89 x 1.40 JC, = 0.70 +11 + $(1.89) = 1.323 ft² = 2.43 ft Y = 1 (140) = 0.467 ft

A = 1:LX 1.40 = 1.54 ft 2 x = 0.70 + L (11) = 1.25 ft I₂ = 1 (1.4) = 0.70 ft z = 0.70-4 (0-10) Az = I (0.702 = 0.7693 ? 3 ft = 0.403 Y = 0.70 ft. A4 = I d² = I (07507² 82 = 1.14 ft 4 = 0.700 t =0.442 ft2 Part A . Axea of the Body = A + A + Az - A4 = 1.323 + 1.54 +0.769 -0.442 A = 3.190ft Part B 7 X = Aise + A + A₃x₃ - A4X4 A + 2 + Az - A4 71 222 x 2.43) + (1.59 x 1.25) +(0.769 X 0.403) -(0.442X1-(4) 3.19

5 = 1.550 ft | Part c 7 J = A, 4, + A₂Y + A3Y - Aage - A+ A₂+ Az - A4 -(1.323 70-467)+(1:54X0.70) +(0.769X0.70) -(0:442 X0700) 3.19 I } = 0.603 ft Part D & Calculate volume of the outer cone Vi = 1/2 198ho (1.80 = 1.25 in) (ho = 4 in) = {(3.4)(1.25)*(4) V = 6.5444 in Simillarly volume of the inner cone :(; = .1017 hi = 3.6 in) V2 = }(314) (1.60)*x 36 = 4.56 in

(8 = 1.50 in) Calculate volume of sphere - V₂ = 4 1982 Vg = 4(3.K) (1.50) V₂ = 14.13 in3 the sphere from the Distance of the Centroid of bottom. hz = ho tr 82 - 8,2 = 4 + V (1:50)2-C1-1033 h = 5.020 in The the the outer cone from Centroid distance for Bottom ho ho h = 3 ho 3 hi = 3 in The Cerroid distance for h₂ = ho-hi the inner cone from the Bottomis 4 = 4-3.6 3.1 in

The Centroid of the Composite Body about 2-axis Z = Vehi - V2 hz + V₂ hz. V - V₂ + V3 Ź -(6.54x3) - (4.56X3.1) +(14.13 X 5.02) 6.54 - 4.56 + 14.13 - 76.4166 16.11 z = 4.743 ft