3Describing Rotational Motion Angular displacement qLet O be the axis of rotationHow far the object has rotatedOnly 2 directions possible: clockwise(-) and counter-clockwise(+)Measured in radians1 radian (rad) is the angle subtended by an arc whose length is equal to the radius of motion
4Distance traveled Arc length traversed For one complete revolution q can be expressed in revolutions
5Example: Bike WheelA bike wheel rotates 4.5 revolutions. How many radians has it rotated?If the wheel has a diameter of 45 cm, what is the distance traveled by a point on the rim of the wheel?
6Example: Bird of PreyA bird’s eye can distinguish objects that subtend an angle no smaller than 3 10-4 rad. How many degrees is this?How small an object can the bird just distinguish when flying at a height of 100m?For small angles (<15), arc length and chord length are nearly the same
7Angular Velocity w Average w Instantaneous w Dt must be very smallVelocity v of a point on a rotating wheelChanges direction as vector turnsIncreases in proportion to distance from the axis of rotation
8Angular Acceleration a Average aInstantaneous aMake Dt as small as possibleTangential accelerationRadial acceleration
9Review of Linear and Angular Quantities Frequency = number of complete revolutions per second = fw = 2pfPeriod = time required to complete one revolution = T = 1/f
10Equations of Motion Zero angular acceleration a = 0, w = constantUniform circular motionq = wt + qoLinear velocity is not constantMagnitude is constant: v = wrDirection is changingAcceleration is not constantatan= 0 but aR= rw2 = constant (centripetal)
11Example: Earth’s Rotation How fast is the earth’s equator turning?w = 2p/T = (2p rad)/84,600s = 7.27 x 10-5 rad/sv = rw = (6,380 km)(7.27 x 10-5 rad/s) = 464 m/sHow will your speed change as you go to the North or South pole?v = (r cos f)w = (464 cos f) m/sf = 14.5°, v = 449 m/sf = 30°, v = 402 m/sf = 60°, v = 232 m/sf = 90°, v = 0 m/s
12The Coriolis EffectAs you go from the equator towards the N pole, you are moving faster than the ground you are moving into: veer to your right (earth rotates west to east)As you go from the N pole towards the equator, you are moving slower than the ground you are moving into: veer to your rightClockwise flow!To or from the S pole: veer to the left!Counterclockwise flow!
13Example: Hard DriveThe platter of the hard drive of a computer rotates at 7200 rpm. What is the angular velocity of the platter?If the reading head of the drive is 3.00 cm from the axis of rotation, how fast is the disk moving right under the head?
14Example (continued)If a single bit requires 0.50 mm of length along the direction of motion, how many bits per second can the writing head write when it is 3.00 cm from the axis?The number of bits passing the head per second isor 45 megabits/s (Mbps)
15Constant Angular Acceleration a = constantw = wo + atq = qo + wot + ½ at2Eliminate t between w and qw2 = wo2 + 2aq
16Total Acceleration atotal = atan + aR atan aR Constant magnitude, changing directionaRVariable magnitude, variable direction
17Example: CentrifugeA centrifuge motor is accelerated from rest to 20,000 rpm in 30s. Determine its angular acceleration and how many revolutions it makes while it is accelerating.SolutionAssuming constant angular acceleration
18Example continued Where the final angular velocity w is The angular displacement in 30s is thenWe divide by 2p to convert to revolutions
19Rolling Motion Translational + rotational motion No Slipping Static friction between object and rolling surface
20Example: BicycleA bicycle slows down uniformly from a velocity of 8.40 m/s to rest over a distance of 115 m. The overall diameter of the tire is 68.0 cm. Determine the initial angular velocity of the wheels.
21Example (continued)Determine the number of revolutions each wheel undergoes before stopping.The rim of the wheel turns 115m before stopping. Thus,Determine the angular acceleration of the wheel
22Example continued Determine the time it took the bicycle to stop Note: when the bike tire completes one revolution, the bike advances a distance equal to the outer circumference of the tire (no slipping or sliding).
23Announcements FINAL EXAM Long Test 4 Wednesday, March 19 7.30 - 10.30 Thursday, March 136.00 – 7.30Room TBA c/o Paulo
24Center of MassYou can reduce an object to a point and describe its translational motion by considering the motion of this point (called its center of mass)
25Determining Center of Mass Consider masses m1, m2, m3, … with coordinates (x1, y1), (x2, y2), (x3, y3), …
26CM for a LegDetermine the center of mass of a leg when a) stretched out and b) bent at 90°. Assume the person is 1.70 m tall.Solutiona) Straight legEssentially 1-DMeasure distance from hip jointCM is = 31.7 units from base of footFor a height of 172 cm, xcm = 54.5 cm above the bottom of the footunits
27CM of Leg b) Bent leg For a height of 172 cm xcm = (172 cm)(0.149) = 25.6 cmycm = (172 cm)(0.23) = 39.6 cmCenter of mass of bent leg is 39.6 cm above the floor and 25.6 cm from the hip joint!unitsunits
28CM TrajectoryCenter of mass of swimmer in flight follows projectile motion (parabolic) trajectoryCenter of mass of wrench follows constant velocity trajectory
29TorqueWhat causes an object to rotate?Torque = force x lever arm
30More Torque Units: Nm (Newton-meter) Torque is a vector quantity Reserve J for work and energyTorque is a vector quantityDirection determined by the right hand rule
31Newton’s First law Translational Equilibrium Rotational Equilibrium All forces cancel out: SF = 0Rotational EquilibriumTorques must balance out: SG = 0When is an object in equilibrium?
32Newton’s Second Law F = ma G = Ia I = moment of inertia a = angular accelerationOnly two possible directionsCounter-clockwise rotationClockwise rotation
33Moment of Inertia of Particles For a single moving object with mass mt = rF = rma = rmra =mr2aI = mr2For several objects rigidly attached to each otherSt = (Smiri2)aI = Smiri2
34Changing Moment of Inertia Determine the change in the moment of inertia of a particle as the radius of its orbit doublesSolutionIt increases by
35Changing Your I Vertical axis of rotation Arms on the side R = 25 cm, M = 9.6 kgRaise your arms in a crucifixion poseR = 57.5 cm, M = 9.6 kg433% increase
38Example: Ball Rolling Down an Inclined Plane Determine the speed of a solid sphere of mass M and radius R when it reaches the bottom of an inclined plane if it starts from rest at a height H and rolls without slipping. Assume no slipping occurs. Compare the result to an object of the same mass sliding down a frictionless inclined plane.
39Solution Initial mechanical energy Final mechanical energy PE = MgHKEtrans = 0KErot = 0Final mechanical energyPE = 0KEtrans = ½ Mv2KErot = ½ Iw2Conservation of EnergyMgH = ½ (Mv2 + Iw2)
40Solution (continued)I = (2/5)MR2 for a solid sphere rotating about an axis through its center of massw = v/RThusMgH = ½ Mv2 + ½(2/5)MR2(v/R)2(1/2 + 1/5) v2 = gHv = [(10/7)gH]1/2v does not depend on the mass and radius of the sphere!!
41Frictionless Incline Ball slides down the incline and does not roll Thus,½ Mv2 = MgHv = (2gH)1/2The speed is greater!None of the original PE is converted into rotational energy.
42Work Done on a Rotating Body W = FDl = F rDqW = tDqPowerP = W/DtP = tDq/Dt = tw
43Angular Momentum LL = IwNewton’s second law becomesThus,
44Conservation of Angular Momentum If the net torque acting on a rotating object is zero, then its angular momentum remains constant.
45The Ice SkaterHow can the ice skater spin so fast?
46The Diver How can the diver make somersaults? Does she have to rotate initially?What trajectory does she follow?
47The Hanging Wheel Why is the wheel standing up? Why does it turn around about the point of support?
48Rotating Disk DemoWhat happens when you tilt the rotating disk?HINT:
49Drunk Driver Test/Tightrope Artist Follow the line walkIncrease your moment of inertia to minimize rotations
50Quiz 81. A 4 kg mass sits at the origin, and a 10 kg mass sits at x = + 21 m. Where is the center of mass on the x-axis? (a) + 7 m (b) m (c) + 14 m (d) + 15 m 2. An object moving in a circular path experiences (a) free fall. (b) constant acceleration. (c) linear acceleration. (d) centripetal acceleration. 3. A boy and a girl are riding on a merry-go-round which is turning at a constant rate. The boy is near the outer edge, and the girl is closer to the center. Who has the greater angular velocity? a) The boy b) The girl c) Both have the same non-zero angular velocity. d) Both have zero angular velocity.
51Quiz 84. A wheel starts at rest, and has an angular acceleration of 4 rad/s2. Through what angle does it turn in 3 s? a) 36 rad b) 18 rad c) 12 rad d) 9 rad 5. A wheel of diameter 26 cm turns at 1500 rpm. How far will a point on the outer rim move in 2 s? a) 314 cm b) 4084 cm c) cm d) cm 6. What is the centripetal acceleration of a point on the perimeter of a bicycle wheel of diameter 70 cm when the bike is moving 8 m/s? a) 91 m/s2 b) 183 m/s2 c) 206 m/s2 d) 266 m/s2 7. A bicycle is moving 4 m/s. What is the angular speed of a wheel if its radius is 30 cm? a) 0.36 rad/s b) 1.2 rad/s c) 4.8 rad/s d) 13.3 rad/s
52Quiz 88. An ice skater is in a spin with his arms outstretched. If he pulls in his arms, what happens to his kinetic energy? a) It increases. b) It decreases. c) It remains constant but non-zero. d) It remains zero. 9. What is the quantity used to measure an object's resistance to changes in rotation? a) mass b) moment of inertia c) linear momentum d) angular momentum 10. A wheel of moment of inertia of 5.00 kg-m2 starts from rest and accelerates under a constant torque of 3.00 N-m for 8.00 s. What is the wheel's rotational kinetic energy at the end of 8.00 s? a) 57.6 J b) 64.0 J c) 78.8 J d) 122 J