PHYS 131 - CHAPTER10:ROTATIONOFARIGIDOBJECTABOUTFIXEDAXIS

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CHAPTER 10: ROTATION OF A RIGID OBJECT ABOUT FIXED AXIS Arc Length,  s  –  distance traveled by particle moving along circular          path   of radius  r . As particle moves along arc, radius line sweeps       out an  angle  θ  = s/r Angular Displacement θ   in rad     θ (rad) Π  /180 * θ (deg) Average Angular Speed ϖ avg  =   θ f  –      θ    i   =   ∆θ    in Rad/s          ϖ avg  =  ½( ϖ f + ϖ i )             t f  – t i          t Instantaneous Angular Speed ϖ lim t-->0  =  ∆θ / t = d θ /dt Average Angular Acceleration α avg  =   ϖ f  –      ϖ    i   ϖ                                                   t f  – t i          t Instantaneous Angular Acceleration,   α lim t-->0  =  ∆ϖ / t=d ϖ /dt Tangential Speed:  v = r ϖ      ϖ  = v/r   Tangential Acceleration: every pt on object – same  ϖ α                 a t  = r α   but  t  depend on radial distance from axis Centripetal Acceleration:    a c  = v 2 /r = r ϖ 2     a r  = a c   Total Linear Acceleration: a=sqrt(a t 2  + a r 2 )=r sqrt( α 2 + ϖ 4 ) Equations of Rotational Kinematics    (Apply when body rotates about a fixed axis)   ϖ f   =  ϖ i  +  α t                    ϖ f 2  =  ϖ i 2  + 2 α ( θ f  –  θ i )   θ f  =  θ I  +  ϖ t + ½ α t 2         θ f  =  θ I  + ½ α t 2   Moments of Inertia of Homogeneous Rigid Objects   Units for I: kg m 2 Hoop / Thin Cylindrical Shell: I cm = MR 2 Hollow Cylinder : I cm = M(R 1 2 + R 2 2 ) Solid Cylinder / Disk: I cm = ½MR 2 Rectangular Plate: I cm = 1/12 M(a 2 +b 2 ) Long Thin Rod w/ Rotation Axis Thru Center: I cm = 1/12 ML 2 Long Thin Rod w/ Rotation Axis Thru End: I cm = 1/3 ML 2 Solid Sphere: I cm =2/5 MR 2 Thin Spherical Shell: I cm =2/3 MR 2 Moment of Inertia of Extended Rigid Object:   r 2 dm  dm=  ρ dV Parallel Axis Theorem : I=I cm +MD 2 Ex. Uniform rigid rod rotating about axis parallel to axis passing thru CM.       y axis            CM      Length of rod: L                                 I = 1/12 ML 2 + M(L/2) 2   I =  ½MR 2   +  M(R/2) 2   = I cm Ex. Particle attached to Meter Stick at 100cm mark. Find L when rotating about axis perpendicular to horizontal table at 50 cm mark & at 0cm mark. 50cm–Center Mass: 2 masses– Meter Stick
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PHYS 131 - CHAPTER10:ROTATIONOFARIGIDOBJECTABOUTFIXEDAXIS

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