ΣF = ma Στ = Iα ½mv 2 ½Iω 2. mv Iω

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Karin Alexander
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1 Thur Oct 22 Assign 9 Friday Today: Torques Angular Momentum x θ v ω a α F τ m I Roll without slipping: x = r Δθ v LINEAR = r ω a LINEAR = r α ΣF = ma Στ = Iα ½mv 2 ½Iω 2 I POINT = MR 2 I HOOP = MR 2 I CYL = ½MR 2 I SPHERE = (2/5)MR 2 (Copy these down in notes) mv Iω

2 Moment of Inertia "An object's moment of inertia determines how much it resists rotational motion. In general for a given total mass, the more mass is concentrated near the axis of rotation, the lower the moment of inertia is Hollow sphere versus solid sphere Hoop versus solid cylinder Meter stick spinning around end or center

3 Moment of Inertia Multiple or Compound Objects Add! (or integrate) (just like would add mass)

4 Multiple Objects Add moments of Inertia For example, consider the following: moment of inertia of disk plus moment of inertia of two point particles. This is all spinning about the center of the disk I TOTAL = I cylinder + I POINTA + I POINTB I TOTAL = 1/2 M CYL R 2 + M A R A 2 + M B R B 2 R A R B

5 Two forces are exerted on a wheel which has a fixed axle at the center. Force A is applied at the rim. Force B is applied halfway between the axle and the rim. F A = ½ F B Which best describes the direction of the angular acceleration? 1. Counterclockwise F B 2. Clockwise 3. Zero F A F A is trying to twist CCW, F B is trying to twist CW. The torques are the same. Torque from A has half the force, but twice the lever arm.

6 A 0.50kg mass is hung from a massive, frictionless pulley of mass 1.5kg and radius 0.10m. Starting from rest, how long will it take for the mass to fall 1.0 m? A bit like: Block A applies resistance through inertia

7 Each Red dot represents a 1kg mass on a turntable. Which of the three turntables requires the least torque to get it from rest to an angular speed of 3 rad/s over 10 s? (1) (2) (3) (4) All the same τ = I α All three same angular acceleration α Smallest moment of inertia requires least torque Masses closest to the center smallest Moment of Inertia If needed to calculate I: I TOTAL = 1/2 M CYL R 2 + M A R A 2 + M B R B 2 + M C R C 2

8 Consider two masses on a rod rotating as shown. In case A, the masses are at the end of the rod. In case B the same 2 masses are closer to the center. If the same torque is applied to each situation, which arrangement of masses has the lowest angular acceleration? 1. A 2. B 3. Both the same A B τ = I α Largest moment of Inertia, lowest angular acceleration (same torque)

9 Consider two masses on a rod rotating as shown. The masses are placed identically in each situation. In each case the rod (presume these to be massless) is turned by a string wrapped around the center axle. The center axle in B has a much larger radius. If the force is the same, which situation has the smallest angular acceleration? 1. A 2. B 3. Both the same A F B F τ = I α Same moment of Inertia; smaller torque (smaller lever arm) => smaller angular acceleration

10 Consider a post which rotates in a frictionless bearing. Two masses are placed on an arm which spins. A string is wrapped around the center post, run over a pulley and attached to a weight. How much time will it take for the weight to fall? τ = I α v LINEAR = r ω a LINEAR = r α

11 We apply the same net torque to 3 objects. Which experiences the greatest angular acceleration? 1. (A) Hoop of mass m and radius r 2. (B) Solid Cylinder of mass m and radius r 3. (C) Hoop of mass ½m and radius r 4. A and B 5. B and C 6. A and C I A = MR 2 = mr 2 7. All I B = ½MR 2 = ½mr 2 I C = MR 2 = (½m)r 2 Στ = Iα Same Torque; Smallest I => Largest α

12 Conservation of Angular Momentum Angular Momentum = L = Iω Stays same unless non-zero net external torque L INITIAL = L FINAL I I ω I = I F ω F Same magnitude AND DIRECTION: Top gyroscope can be used for navigation football good spiral

13 A student stands on a turntable holding two 2-kg masses in their hands. Their arms are folded close to their body. They are rotating. When the student stretches their arms out (with the masses in hand), the angular velocity of the student will: 1. Increase 2. Decrease 3. Stay the Same L INITIAL = L FINAL I I ω I = I F ω F Moment of Inertia increases, angular speed decreases

14 Rotational Variables x θ v ω a α F τ m I ΣF = ma Στ = Iα ½mv 2 ½Iω 2 mv Iω I POINT = MR 2 I HOOP = MR 2 I CYL = ½MR 2 I SPHERE = (2/5)MR 2

15 Each Red dot represents a 1kg mass on a massless turntable. The inner circle has a radius half that of the outer circle. If the masses start in Configuration A and move to Configuration B while the turntable is spinning, how does the final angular momentum of the turntable compare to the initial? (assume no external torque) (A) (B) 1. Greater 2. Smaller 3. Same Angular Momentum Conserved L I =L F

16 Each Red dot represents a 1kg mass on a massless turntable. The inner circle has a radius half that of the outer circle. If the masses start in Configuration A and move to Configuration B while the turntable is spinning, how does the final angular speed of the turntable compare to the initial? (assume no external torque) (A) (B) 1. Greater 2. Smaller 3. Same Angular Momentum stays same Moment of Inertia is smaller I A = (1kg)R 2 +(1kg)R 2 + (1kg)(½R) 2 I B = (1kg)R 2 +(1kg)(½R) 2 + (1kg)(½R) 2 L INITIAL = L FINAL I A ω A = I B ω B I A > I B => ω A < ω B

17 Suppose a rod of negligible mass has a 2.0-kg object on each end. The rod is 1.0m long. The rod is spinning at an angular speed of 5.0rad/s. A small motor on the rod now pulls the objects in such that they are 0.25 m from the axis of rotation. There are no external torques on the system. What is the new angular speed? L i = L f I i ω i = I f ω f (MR i 2 +MR i2 )ω i = (MR f 2 +MR f2 )ω f 2(2.0kg)(0.50) 2 (5.0rad/s) = 2(2.0kg)(0.25) 2 ω f 5.0kg m 2 /s=0.25 ω f ω f = 5.0/0.25 = 20. rad/s

18 Carousel Example Playground carousel As people redistribute, moment of inertia changes As moment of inertia changes, angular speed changes

19 Conservation of Angular Momentum in Solar System Objects in orbit follow elliptical orbit. Move faster when closer - R is smaller, so I is smaller, ω bigger: L I = L F I I ω I = I F ω F MR I 2 ω I =MR F 2 ω F

20 Conservation of Energy Total Mechanical Energy = KE + PE = (KE TRANS + KE ROT ) + PE KE ROT = ½Iω 2 If rotating, has KE ROT If W NC = 0, (KE TRANS +KE ROT + PE) INIT = (KE TRANS + KE ROT + PE) FINAL Not responsible for KE ROT for quiz and exam

21 A solid cylinder and a hoop roll down an incline. Both are the same mass and radius. Which reaches the bottom of the incline first? 1. The cylinder 2. The hoop 3. Both at same time "An object's moment of inertia determines how much it resists rotational motion. The cylinder has a smaller moment of inertia, so it s easier to get going. Less energy goes into rolling the cylinder, so it has more translational kinetic energy at the bottom. Turns out that mass and radius don t matter solid cylinders will always beat hoops.

Rolling Racers: four objects with same mass are placed on a ramp and left to roll without slipping. Starting from rest, which one is traveling the ramp length faster? 1.

22 Rolling Racers: four objects with same mass are placed on a ramp and left to roll without slipping. Starting from rest, which one is traveling the ramp length faster? 1. Spherical shell (I=2/3 MR 2 ) 2. Solid sphere (I=2/5 MR 2 ) 3. Hollow cylinder (I=MR 2 ) 4. Solid cylinder (I=1/2 MR 2 ) The solid sphere has a smaller moment of inertia, so it s easier to get going.

23 Each Red dot represents a 1kg mass on a massless turntable. The blue dot is 2kg. The inner circle has a radius half that of the outer circle. If the masses start in Configuration A and move to Configuration B while the turntable is spinning, how does the final angular speed of the turntable compare to the initial? (assume no external torque) (A) (B) 1. Greater 2. Smaller 3. Same L INITIAL = L FINAL Moment of Inertia stays same I A = (1kg)R 2 +(1kg)R 2 + (2kg)(½R) 2 I B = (2kg)R 2 +(1kg)(½R) 2 + (1kg)(½R) 2 I A ω A = I B ω B I A = I B => ω A = ω B

24 A wheel of radius 0.30m rolls without slipping 15m along the sidewalk. How many revolutions did the the wheel undergo? (1) 0.72 rev (2) 6.2 rev (3) 8.0 rev (4) 9.7 rev (5) 15 rev (6) 50 rev Rolling w/o slipping x = r θ 15 m =0.3 θ θ = 50 rad 50 rad (1 rev/2π rad) = 7.96 rev Or Circumference = 2π r = 1.885m Distance/Circumference = # rev 15m/ m = 7.96 rev

25 Pulsars Supernova remants Star collapses into VERY dense object neutron star Typical radius about 10km, but typical mass 1.5 times mass of Sun Teaspoon of neutron star material would weight 1 billion tons. Spinning pretty quickly, especially for such a large object Huge magnetic fields

Chapter 9-10 Test Review

Chapter 9-10 Test Review Chapter Summary 9.2. The Second Condition for Equilibrium Explain torque and the factors on which it depends. Describe the role of torque in rotational mechanics. 10.1. Angular