Lecture 18: Angular Acceleration & Angular MomentumPowerPoint PresentationSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Lecture 18: Angular Acceleration &Angular MomentumQuestions of Yesterday1) If an object is rotating at a constant angular speed which statement is true?a) the system is in equilibriumb) the net force on the object is ZEROc) the net torque on the object is ZEROd) all of the above2) Student 1 (mass = m) sits on the left end of massless seesaw of length L and Student 2 (mass = 2m) sits at the right end. Where must the pivot be placed so the system is in equilibrium?a) L/2 b) L/3 from the right (from Student 2)c) L/3 from the left (from Student 1)d) the system cant be in equilibriumTorque & Angular AccelerationForce causes linear accelerationTORQUE causes angular accelerationA system is in equilibrium (a = 0,  = 0) when ∑F = 0 and ∑ = 0What is the induced angular acceleration  of an object due to a certain torque  acting on an object? aT=rF = ma = ?  = rFsinTorque & Angular AccelerationrFTmFT = maTFTr = mraTaT=r = r*Fsin = mr2Moment of Inertia (I) of a mass m rotating about an axis at a distance r from the axis = mr2mr2 = Moment of Inertia (I)Moment of Inertiar2FTm1m2r1 = 1 + 2… = m1r121 + m2r222… = mr2 = (m1r12 + m2r22…) = (mr2) = II = m1r12 + m2r22… = mr2 1 = 2 = Moment of Inertia depends on axis of rotation!!!Moment of Inertia of a Rigid Body = ImmMass is distributed over entire body (from 0 to r)I = m1r12 + m2r22… =  mr2 Angular acceleration of every point on rigid body is equalWhat is the Moment of Inertia of a rotating rigid body?Moment of Inertia of a Rigid BodyWhat is the Moment of Inertia of a rotating rigid body?m = II = mr2 I1 = m1r12I2 = m2r22 = m1r121 + m2r222… = (mr2)Moment of Inertia of a rigid body depends on the MASS of the object AND the DISTRIBUTION of mass about the AXIS of rotationMoment of Inertia of a Rigid BodyWhich object has a greater Moment of Inertia?I = mr2 rm mrIf the same force F is applied to each object as shown…which object will have a greater angular acceleration?F FMoment of Inertia of a Rigid BodyI = mr2 In which case is the moment of inertia of the baton greater?m mLaxis of rotationm mLaxis of rotationIf both batons were rotating with the same , and the same braking torque is applied to both…Which one would come to rest sooner?Rotational Kinetic EnergyWhat is the kinetic energy of a rotating object?r1mI1 = m1r12I2 = m2r22rvmKE = (1/2)mv2KE = (1/2)m(r)2KEr = (1/2)I2KEr = KEr1 + KEr2…= ∑KErKEr = ∑(1/2)mv2KEr = ∑(1/2)mr22KEr = (1/2)(∑mr2 )2KEr = (1/2)I2Is energy conserved as the ball rolls down the frictionless ramp?Rotational Kinetic EnergyConservation of Mechanical Energy when Wnc = 0(KEt + KEr + PEG + PES)i = (KEt + KEr + PEG + PES)f(1/2)mv2mgy (1/2)I2(1/2)kx2What forms of energy does the ball have while rolling down the ramp?Rotational Kinetic Energy(1/2)mv2(1/2)I2(KEt + KEr + PEG + PES)i = (KEt + KEr + PEG + PES)fmgy(1/2)kx2Conservation of Mechanical Energy when Wnc = 0Work-Energy TheoremWnc = KEt + KEr + PEG + PESWhat forms of energy does each object have…. at the top of the ramp (before being released)?halfway down the ramp?at the bottom of the ramp?Rotational Kinetic Energyhh What is the speed of each object when it reaches the bottom of the frictionless ramp (in terms of m,g, h, R and )?Which object reaches the bottom first?mmSphere radius = RIs = (2/5)mR2Cube length = 2RThe net TORQUE acting on an object is equal to the CHANGE in ANGULAR MOMENTUM in a given TIME intervalAngular Momentump = mvLinear Momentum=ptRelating F & p∑F∑F = mamv)t=Newton’s 2nd LawRotational Analog toNewton’s 2nd Law∑ = II)t=Angular MomentumL = IRelating  & L=Lt∑The net TORQUE acting on an object is equal to the CHANGE in ANGULAR MOMENTUM in a given TIME intervalConservation of Angular MomentumIf NO net external TORQUE is acting on an object then ANGULAR MOMENTUM is CONSERVED=Lt∑if ∑= 0 then L = 0 Li = LfIii= Iff if ∑= 0Conservation of Angular MomentumYou (mass m) are standing at the center of a merry-go-round (I = (1/2)MR2) which is rotating with angular speed 1, as you walk to the outer edge of the merry-go-round…What happens the angular momentum of the system?What happens the angular speed of the merry-go-round?What happens to the rotational kinetic energy of the system? Angular MomentumRMRM12 = ?A 10.00-kg cylindrical reel with the radius of 0.500 m and a frictionless axle starts from rest and speeds up uniformly as a 5.00 kg bucket falls into a well, making a light rope unwind from the reel. The bucket starts from rest and falls for 5.00 s.Practice ProblemWhat is the linear acceleration of the falling bucket?How far does it drop?What is the angular acceleration of the reel?Use energy conservation principles to determine the speed of the spool after the bucket has fallen 5.00 m5.00 kg10.0 kg0.500 mTwo astronauts, each having a mass of 100.0-kg, are connected by a 10.0 m rope of negligible mass. They are isolated in space, moving in circles around the point halfway between them at a speed of 5.00 m/s. Treating the astronauts as particles…What is the magnitude of the angular momentum and the rotational energy of the system?By pulling on the rope, the astronauts shorten the distance between them to 5.00 m…What is the new angular momentum of the system?What are their new angular and linear speeds?What is the new rotational energy of the system?How much work is done by the astronauts in shortening the rope?Practice ProblemQuestions of the Day1) A solid sphere and a hoop of equal radius and mass are both rolled up an incline with the same initial velocity. Which object will travel farthest up the inclined plane? a) the sphereb) the hoopc) they’ll both travel the same distance up the planed) it depends on the angle of the incline2) If an acrobat rotates once each second while sailing through the air, and then contracts to reduce her moment of inertia to 1/3 of what is was, how many rotations per second will result? a) once each second b) 3 times each secondc) 1/3 times each secondd) 9 times