Rotational EnergyRigid BodyTranslation and RotationRotational MotionCompact DiscCircular EnergyIntegrating MassSpinning EarthRotational EnergyRotational EnergyRigid BodyRigid BodyReal objects have mass at Real objects have mass at points other than the center points other than the center of mass.of mass.Each point in an object can Each point in an object can be measured from an origin be measured from an origin at the center of mass.at the center of mass.If the positions are fixed If the positions are fixed compared to the center of compared to the center of mass it is a mass it is a rigid bodyrigid body..riTranslation and RotationTranslation and RotationThe motion of a rigid body The motion of a rigid body includes the motion of its includes the motion of its center of mass.center of mass.This is This is translationaltranslational motion motionA rigid body can also move A rigid body can also move while its center of mass is while its center of mass is fixed.fixed.This is This is rotationalrotational motion. motion.vCMRotational MotionRotational MotionKinematic equations with Kinematic equations with constant linear acceleration constant linear acceleration were defined.were defined.•vv = = vv00 + + atat•xx = = xx00 + + vv00 tt + ½ + ½atat22•vv22 = = vv00 22 + 2 + 2aa((xx - - xx00 ) )Kinematic equations with Kinematic equations with constant angular constant angular acceleration are similar.acceleration are similar.•  = = 00 + + tt•  = = 00 + + 00 tt + ½ + ½tt22• 22 = = 00 22 + 2 + 2 (( - - 00 ) )Compact DiscCompact DiscA CD spins so that the A CD spins so that the tangential speed is constant.tangential speed is constant.•The constant speed is 1.3 m/sThe constant speed is 1.3 m/s•The inner radius is at 0.023 mThe inner radius is at 0.023 m•The outer radius is at 0.058 mThe outer radius is at 0.058 m•The total time is 74 min, 33 s = The total time is 74 min, 33 s = 4500 s4500 sFind the angular velocity at the Find the angular velocity at the beginning (inner) and end.beginning (inner) and end.Find the constant angular Find the constant angular acceleration.acceleration.Angular velocity is Angular velocity is  == vv//rr..•Inner: Inner:  = 1.3 m/s / 0.023 m = 1.3 m/s / 0.023 m == 57 rad/s.57 rad/s.•Outer: Outer:  = 1.3 m/s /0.058 m = 1.3 m/s /0.058 m == 22 rad/s.22 rad/s.Angular acceleration isAngular acceleration is23srad108.7srad/s45005722ttififCircular EnergyCircular EnergyObjects in circular motion have kinetic energy.Objects in circular motion have kinetic energy.•KK = ½ = ½ mm v v22The velocity can be converted to angular quantities.The velocity can be converted to angular quantities.•KK = ½ = ½ mm ( (rr ))22•KK = ½ ( = ½ (mm rr22) ) 22The term The term ((mm rr22) ) is the moment of inertia of a particle.is the moment of inertia of a particle.Integrating Mass Integrating Mass 22122212221221)())((IKdmrdKKdmrdKrdmdKrotrotThe kinetic energy is due to The kinetic energy is due to the kinetic energy of the the kinetic energy of the individual pieces.individual pieces.The form is similar to linear The form is similar to linear kinetic energy.kinetic energy.•KKCMCM = ½ = ½ mm vv22•KKrotrot = ½ = ½ II 22Spinning EarthSpinning EarthHow much energy is stored How much energy is stored in the spinning earth?in the spinning earth?The earth spins about its The earth spins about its axis.axis.•The moment of inertia for a The moment of inertia for a sphere: sphere: II = 2/5 = 2/5 MM RR22•The kinetic energy for the The kinetic energy for the earth: earth: KKrotrot = 1/5 = 1/5 MM RR22 22•With values: With values: K = 2.56 x 10K = 2.56 x 102929 J JThe energy is equivalent to 250 million times the world’s nuclear