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Fowles08 - 1 CHAPTER 8 MECHANICS OF RIGID BODIES: PLANAR...

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1 CHAPTER 8 MECHANICS OF RIGID BODIES: PLANAR MOTION 8.1 (a) For each portion of the wire having a mass 3 m and centered at , 22 bb    , ( ) 0,0 , and , y x b b 1 00 23 cm bm x m    = −+ +       = 1 0 cm b y m 3 = ++ = (b) () 1 2 ds xdy b y dy ==− 1 2 0 1 b cm yy b y m ρ =− d y 1 2 0 2 2 1 4 yb y cm byd by y b πρ = = −− − = 4 y 3 cm b π = From symmetry, 4 x 3 cm b = (c) The center of mass is on the y-axis. 1 2 ds xdy by dy == 3 1 2 2 11 0 0 2 2 cm b b y d y yd y y by dy y ∫∫ 3 y = 5 cm b (d) The center of mass is on the z-axis. ( ) 2 x y dz bzdz += dv r dz ππ 1
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2 2 00 bb cm zb z d z zd z z bzdz zdz ρπ == 2 3 cm = (e) The center of mass is on the z-axis. α is the half-angle of the apex of the cone. is the radius of the base at z = 0 and is radius of a circle at some arbitrary z in a plane parallel to the base. r D r tan rr bbz D , a constant () 2 22 tan dv r dz b z dz π πα 23 11 tan 33 2 mr ρ ππρ D 2 2 3 0 3 0 32 tan 3 2 1 tan 3 b b cm zbz d z b b πρ z b z z d z + 4 cm b z = 8.2 2 0 0 b cm b cx dx xdx x dx cxdx 2 3 cm b x = 8.3 The center of mass is on the z-axis. Consider the sphere with the cavity to be made of a (i) solid sphere of radius and mass a s M , with its center of mass at , and (ii) a solid sphere the size of the cavity, with mass 0 z = c M and center of mass at 2 a z = − . The actual sphere with the cavity has a mass s c M mM = and center of mass . cm z 1 0 s 2 cc m a M mz   =− +     M 3 4 3 s M a = , 3 4 c a M = 3 3 1 0 2 cm aa a az a   +   2
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3 14 cm a z = 8.4 (a) 22 2 0 32 2 zi i i mb b R   == + +     Im 2 6 z mb I = (b) ds rd dr θ = , sin Rr = 2 z I Rd s ρ = 4 0 4 sin rb z r I rr d r d π θθ == = =− = ∫∫ 4 2 4 4 sin 4 z b I d = 2 sin 2 sin 24 d 4 1 442 z b I ρπ 2 1 4 = () 2 2 4 z mb I (c) 2 x ds hdx b dx b Where the parabola intersects the line , yb = 1 2 x by b ± bb y xx I x b dx bx dx ρρ −− 4 4 15 y I b = 2 2 4 3 b b x d x b b =−= 2 1 5 y I mb = (d) dv 2 RhdR = hbz = 3
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4 () 1 1 22 2 2 R xy b z =+ = 1 2 1 2 b dR dz z  =   1 1 2 2 2 0 1 2 2 b z b I Rd v b z b z bz d z z ρρπ == ∫∫ 0 1 6 b z 5 I bb z z d z b π ρπ =− = ρ 1 1 2 2 0 1 2 2 b b md v b z b z z d z 3 0 1 2 b mb b z d z b = 2 1 3 z I mb = (e) α is the half-angle of the apex of the cone. r is the radius of the base at z = 0 and is radius of a circle at some arbitrary z in a plane parallel to the base. D r , a constant tan R R z D bz dv RhdR ππ R zdR b D Since R = , , and the limits of integration for correspond to zb R b D R dR dz b D 0 R R =→ D 0 2 2 0 2 2 2 z bzR I v z −− DD R b 4 32 23 4 4 1 3 b z R 4 0 10 I b z z d z πρ =+ + = D Rb b D d z b D 2 1 3 mR b D = 2 3 10 z I mR D = 4
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5 8.5 Consider the sphere with the cavity to be made of a (i) solid sphere of radius and and mass a s M , with its center of mass at 0 z = , and (ii) a solid sphere the size of the cavity, with mass c M and center of mass at 2 a z = − .
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This note was uploaded on 03/09/2008 for the course PHYS 301 taught by Professor Mokhtari during the Fall '04 term at UCLA.

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Fowles08 - 1 CHAPTER 8 MECHANICS OF RIGID BODIES: PLANAR...

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