Question

11-4. The assembly shown below is comprised of a steel hemisphere and an aluminum cylinder with densities, Pst = 7.50 Mg/m and pal = 2.5 Mg/m3, respectively. If h 220 mm, (a) Determine the location of the center of mass of the assembly (b) Determine the location of the center of mass of the same assembly (same dimensions) if it was made entirely out of steel. (c) How does the location of the center of mass from part (b) change if the assembly was made entirely out of aluminum rather than steel? (d) In 1-3 sentences, explain why the center of mass found in part (a) is different from the center of mass found in part (b) (e) Find the surface area and volume of this assembly given a homogeneous material (combined hemisphere and cylinder), using the Theorems of Pappus and Guldinus. 80 mm h G 160 mm

Answer 1

Write the mass center of the assembly. Σ, ΣΖη m Here, the mass of each segment is m and the centroid of differential element along z axis is ž. (a). Segment m(Mg) = PV ž(m) žm(Mg.m) * )=(0.160)|(7.50) 0.160-(0.160) Steel hemisphere = 0.06434 = 0.1 0.006434 [z (0.22)(0.08)*](2.50) 0.160 + 0,23 2 Aluminium cylinder = 0.01106 0.0754 = 0.27 0.37 0.002862 0.009296 in Equation (I). Substitute 0.009296 Mg.m for žm and 0.0754 Mg form 0.009296 Mg.m 0.0754 Mg =- = 0.123 mx1,000 mm = 123 mm The mass center of the assembly (7) is 123 mm.

Segment mMg) =pV ž(m) žm (Mg.m) ( )+ (0.160)" |(7:50) 0.160- (0.160) Steel hemisphere = 0.06434 = 0.1 0.006434 0.22 [7 (0.22)(0.08)*](7.50) (0. 0.160 +; 0.160+ 2 Steel cylinder = 0.0331 0.09744 = 0.27 0.37 0.008937 0.01537 z =- 0.01537 0.09744 mg mm = 0.158 m x1,000 m = 158 mm The mass center of the assembly (7) is 158 mm.

(c). Segment m(Mg) = PV ž(m) žm (Mg.m) | (0.160)||(2.50) 0.160-(0.160) Aluminium hemisphere = 0.02145 = 0.1 0.002145 [2 (0.22)(0.08)](2.50) 0.160 + 0,22 Aluminium cylinder = 0.01106 0.03251 = 0.27 0.37 0.002862 0.005007 ž =- 0.005007 0.03251 Mg mm = 0.154 mx1,000+ m = 154 mm The mass center of the assembly (7) is 154 mm. The difference between part (b) and (c) is 4mm