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Physics 151: Lecture 23 Today’s Agenda

Physics 151: Lecture 23 Today’s Agenda. Topics More on Rolling Motion Ch. 11.1 Angular Momentum Ch. 11.3-5. v ?. M. Example : Rolling Motion. A cylinder is about to roll down an inclined plane. What is its speed at the bottom of the plane ?. Cylinder has radius R. M. M. M. M. M.

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Physics 151: Lecture 23 Today’s Agenda

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  1. Physics 151: Lecture 23Today’s Agenda • Topics • More on Rolling Motion • Ch. 11.1 Angular Momentum Ch. 11.3-5

  2. v ? M Example :Rolling Motion • A cylinder is about to roll down an inclined plane. What is its speed at the bottom of the plane ? Cylinder has radius R M M M M M h M q

  3. Lecture 22, ACT 4aRolling Motion • A race !! Two cylinders are rolled down a ramp. They have the same radius but different masses, M1 > M2. Which wins the race to the bottom ? A) Cylinder 1 B) Cylinder 2 C) It will be a tie M1 M2 h M? q

  4. Lecture 22, ACT 4bRolling Motion • A race !! Two cylinders are rolled down a ramp. They have the same moment of inertia but different radius, R1 > R2. Which wins the race to the bottom ? A) Cylinder 1 B) Cylinder 2 C) It will be a tie R1 R2 animation h M? q

  5. Lecture 22, ACT 4cRolling Motion • A race !! A cylinder and a hoop are rolled down a ramp. They have the same mass and the same radius. Which wins the race to the bottom ? A) Cylinder B) Hoop C) It will be a tie M1 M2 animation h M? q

  6. Remember our roller coaster. Perhaps now we can get the ball to go around the circle without anyone dying. Note: Radius of loop = R Radius of ball = r

  7. 1 2 How high do we have to start the ball ? h h = 2.7 R = (2R + 1/2R) + 2/10 R -> The rolling motion added an extra 2/10 R to the height)

  8. The rotational analogue of force F is torque  • Define the rotational analogue of momentum p to be angular momentum See text: 11.3 p=mv Angular Momentum:Definitions & Derivations • We have shown that for a system of particles Momentum is conserved if • What is the rotational version of this ?? Animation

  9. So (so what...?) See text: 11.3 Definitions & Derivations... • First consider the rate of change of L:

  10. See text: 11.3 Definitions & Derivations... • Recall that  EXT • Which finally gives us: • Analogue of !!

  11. In the absence of external torques See text: 11.5 What does it mean? • where and Total angular momentum is conserved

  12. See text: 11.4 Angular momentum of a rigid bodyabout a fixed axis: • Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin is the sum of the angular momenta of each particle: (since ri, vi , are perpendicular) v1 We see that L is in the z direction. m2 j Using vi =  ri, we get r2 m1 r1 i v2  r3 m3 v3 I  Analogue of p = mv !!

  13. Lecture 23, ACT 2Angular momentum • In the figure, a 1.6-kg weight swings in a vertical circle at the end of a string having negligible weight. The string is 2 m long. If the weight is released with zero initial velocity from a horizontal position, its angular momentum (in kg · m2/s) at the lowest point of its path relative to the center of the circle is approximately • a. 40 • b. 10 • c. 30 • d. 20 • e. 50

  14. See text: 11.4 Angular momentum of a rigid bodyabout a fixed axis: • In general, for an object rotating about a fixed (z) axis we can write LZ = I • The direction of LZ is given by theright hand rule (same as ). • We will omit the ”Z” subscript for simplicity,and write L= I z 

  15. K1 > K2 w1 w2 disk 1 disk 2 Lecture 23, ACT 2Angular momentum • Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2. • Which one has the biggest moment of inertia ? (a) disk 1 (b) disk 2 (c) not enough info I1 < I2

  16. z z F Example: Two Disks • A disk of mass M and radius R rotates around the z axis with angular velocity 0. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity F. What is F ? 0

  17. z Example: Two Disks • First realize that there are no external torques acting on the two-disk system. • Angular momentum will be conserved ! • Initially, the total angular momentum is due only to the disk on the bottom: 2 1 0

  18. Example: Two Disks • First realize that there are no external torques acting on the two-disk system. • Angular momentum will be conserved ! • Finally, the total angular momentum is dueto both disks spinning: z 2 1 F

  19. z z F Example: Two Disks • Since LINI = LFIN An inelastic collision, since E is not conserved (friction) ! LINI LFIN 0

  20. z z F Example: Two Disks • Let’s use conservation of energy principle: EINI = EFIN 1/2 Iw02 = 1/2 (I + I)wF2 wF2 = 1/2 w02 wF = w0 / 21/2 EINI EFIN 0

  21. Example: Two Disks • Using conservation of angular momentum: LINI = LFIN we got a different answer ! wF’ = w0 / 21/2 Conservation of energy ! Conservation of momentum ! wF = w0 / 2 wF’ > wF Which one is correct ?

  22. z F Example: Two Disks • Is the system conservative ? • Are there any non-conservative forces involved ? • In order for top disc to turn when in contact with the bottom one there has to be friction ! (non-conservative force !) • So, we can not use the conservation of energy here. • correct answer: wF = w0/2 • We can calculate work being done due to this friction ! W = DE = 1/2 Iw02 - 1/2 (I+I)(w0/2)2 = 1/2 Iw02 (1 - 2/4) = 1/4 Iw02 = 1/8 MR2 w02 This is 1/2 of initial Energy !

  23. Lecture 23, ACT 2Angular momentum

  24. Recap of today’s lecture • Chapter 11.1-5, • Rolling Motion • Angular Momentum • For next time: Read Ch. 11.1-11.

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