Fowles 08 | Torque | Trigonometric Functions

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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 , 2 2 b b | | − | \ . , ( ) 0, 0 , and , 2 2 b b \ . | | | … y x b b 1 0 0 2 3 2 3 cm b m b m x m ( | || | | || | = − + + | | | | ( \ .\ . \ .\ . ¸ ¸ = 1 0 2 3 2 3 cm b m b m b y m ( | || | | || | 3 = + + = | | | | ( \ .\ . \
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   1 CHAPTER 8MECHANICS OF RIGID BODIES:PLANAR MOTION 8.1 (a) For each portion of the wire having a mass3 m and centered at,22 b b  −   , ( ) 0,0, and ,22 b b     … b1002323 cm b m b m xm       = − + +            =  102323 cm b m b m b ym        3 = + + =              (b) ( ) 1222 ds xdy b y dy = = −   ( ) 12220 1 bcm  y y b ym  ρ  = − ∫ dy   ( ) ( ) 12222202 214  y b ycm b y d b y yb  ρ π ρ  == − − −= ∫  4  y 3 cm b π  =  From symmetry,4  x 3 cm b π  =  (c) The center of mass is on the y-axis. ( ) 12 22 ds xdy by dy = =   ( )( ) 312200112200 22 b bcmbb  y by dy y dy yby dyy dy  ρ  ρ  = = ∫ ∫∫∫  3  y =  5 cm b  (d) The center of mass is on the z-axis. ( ) 222  x y dz bzdz  π  + = dv r dz  π π  = =  1   2  20000 b bcmb b  z bzdz z dz  z bzdz zdz   ρ π  ρπ  = = ∫ ∫∫ ∫  23 cm  z b =  (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  ina plane parallel to the base. r  D r  tan r r b b z  α  = =− D , a constant ( ) 222 tan dv r dz b z dz  π π α  = = −   23 11tan33 b b 2 m r   ρ π πρ  = = D α    ( ) ( ) 2222303032 tan321tan3 bbcm  z b z dz  z bbb  ρ π α πρ α  −= = − ∫∫ z bz z dz  +  4 cm b z  =   8.2   200 bcmb cx dx xdx xdxcxdx  ρ  ρ  = = ∫∫∫∫  23 cm b x =   8.3 The center of mass is on the z-axis. Consider the sphere with the cavity to be madeof a (i) solid sphere of radius and mass a  s , with its center of mass at , and (ii) a solid sphere the size of the cavity, withmass0  z  = c − and center of mass at2 a z  = − . The actual sphere withthe cavity has a mass  s c  M m M  = − and center of mass . cm  z  10  s 2 c cm amz    = − +      M    3 43  s a π ρ  = , 3 432 c a M  π ρ   =     3333 10222 cm a a aa z a         = − + −                  2   314 cm a z  =   8.4 (a) 222 0322  z i ii m b b R     = = + +          ∑  I m   2 6  z  mb I  =  (b)  ds rd dr  θ  = , sin  R r  θ  =   2  z   I R ds  ρ  = ∫   22404 sin r b z r   I r rdr d  π θ π θ   ρ θ θ  = ===− = ∫ ∫   4244 sin4  z  b I d  π π   ρ θ θ  − ∫ =   2 sin2sin24 d  θ θ θ θ   = −   ∫   4 1442  z  b I  ρ π   = −     2 14 m b  ρπ  =   ( ) 2 24  z  mb I  π π  = −  (c)  2  xds hdx b dxb  = = −    Where the parabola intersects the line ,  y b = ( ) 12  x by b = = ±   2422 b b yb b  x x I x b dx bx dxb b  ρ ρ  − −    = − = −       ∫ ∫   4 415  y  I b  ρ  =   22 43 bb  xm b dxbb  ρ ρ  −  = − =   ∫   2 15  y  I mb =  (d) dv 2 RhdR π  =   h b z  = −  3   4  ( ) ( ) 112222  R x y bz  = + =   12 12 bdR dz  z   =     ( ) ( ) 112220 122 b z  b I R dv bz bz b z dz  z   ρ ρ π   = = −   ∫ ∫   ( ) 220 16 b z  5  I b bz z dz b πρ π  = − = ∫ ρ    ( ) ( ) 11220 122 b bm dv bz b z  z   ρ ρ π   = = −   ∫ ∫ dz    ( ) 30 12 b m b b z dz b πρ π  = − = ∫ ρ    2 13  z   I mb =  (e) α  is the half-angle of the apex of the cone. r  is theradius of the base at  z  = 0 and is radius of a circle atsome arbitrary  z  in a plane parallel to the base. D r   , a constanttan R Rb b z  α  = =− D   ( ) 22 b z dv RhdR π π  −= = R zdRb D ( )  Since  R = , , and the limits of integration for correspond to  z b   bz Rb − D RdR dz b = − D 0  R R = → D 0 = → ( )   ( ) 22022 2  z  b z R b z R I R dv z   ρ ρ π  − −= = ∫ ∫ D D    R b b b   ( ) 4322344 1233 b z   R 40 10  I b z b z bz z dz  πρ  = + − + − = ∫ D  R bb πρ  D   dz b  −   D   2 13 m R b  ρ π  D =   2 310  z   I mR D =  4
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