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Announcements

Announcements. Quiz #6 is this Friday. It covers Units 10 (center of mass) and 12 (Conservation of momentum: elastic collisions). In-class Exam scores should be available soon. Exam key is now posted on the website as a PDF. Correct answers should also be visible now on Canvas.

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Announcements

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  1. Announcements • Quiz #6 is this Friday. It covers Units 10 (center of mass) and 12 (Conservation of momentum: elastic collisions). • In-class Exam scores should be available soon. • Exam key is now posted on the website as a PDF. • Correct answers should also be visible now on Canvas.

  2. Lecture 14: Rotational Kinematics and Moment of Inertia Today’s Concepts: a) Rotational Motion b) Moment of Inertia

  3. Summary of Rotations Angular velocity ωis measured in radians/sec Frequency f is measured in revolutions/sec 1 revolution =2π radians Period T = 1/f

  4. Another Summary ω Constant αdoes not mean constant ω

  5. ACT: Spinning Disk 1 A disk spins at 2 revolutions/sec. What is its period? A) T= 2 sec B)T= 2π sec C) T=½ sec Period T = 1/f

  6. ACT: Spinning Disk 2 A disk spins at 2 revolutions/sec. What is its angular velocity? A) B) C) rad/sec rad/sec rad/sec

  7. CheckPoint: Spinning wheel α A wheel which is initially at rest starts to turn with a constant angular acceleration. After 4 seconds it has made 4 complete revolutions. How many revolutions has it made after 8 seconds? A) 8 B) 12 C) 16 Less than 40% of you got this correct.

  8. CheckPoint: Discuss and Re-vote α After 4 seconds it has made 4 complete revolutions. How many revolutions has it made after 8 seconds? A) 8 B) 12 C) 16 A) The angular velocity is 1 revolution/ second. 8 seconds, 8 revolutions B) 4 sec for 4 revs and the rate of rotation is increasing because the acceleration is increasing so it increases to 12 in 8 secs C) What goes up, must come down.Spinnin' wheel got ta go round. Talkin' bout your troubles, it's a cryin' sin. Ride a painted pony, let the spinnin' wheel spin. Ya got no money, and ya, ya got no home. Spinnin' wheel, all alone. Talkin' bout your troubles and ya, ya never learn. Ride a painted pony, let the spinnin' wheel turn. In all seriousness though the answer is (C) or 16. theta = (1/2)alpha(t^2), so doubling t ends up quadrupling theta.

  9. Calculating Moment of Inertia Depends on rotation axis

  10. CheckPoint: Triangle of Spheres • A triangular shape is made from identical balls and identical rigid, massless rods as shown. The moment of inertia about the a, b, and c axes is Ia, Ib, and Ic respectively. • Which of the following orderings is correct? About than 46% of you got this correct. A) Ia > Ib > Ic B) Ia> Ic > Ib C) Ib > Ia > Ic a b c

  11. CheckPoint: Discuss and Re-Vote a A) Ia > Ib > Ic B) Ia> Ic > Ib C) Ib > Ia > Ic b c Where riis the perpendicular distance of each mass from the rotation axis

  12. Calculation of Moment of Inertia Bigger when the mass is concentrated further out

  13. Case 1 Case 2 R R ω ω CheckPoint: Rotating Hoop In both cases shown below a hula hoop with mass M and radius R is spun with the same angular velocity about a vertical axis through its center. In Case 1 the plane of the hoop is parallel to the floor and in Case 2 it is perpendicular. In which case does the spinning hoop have the most kinetic energy? A) Case 1 B) Case 2 C) Same Less than 50% of you got this correct.

  14. Case 1 Case 2 R R ω ω CheckPoint Results: Rotating Hoop In which case does the spinning hoop have the most kinetic energy? A) Case 1 B) Case 2 C) Same A) Since the angular velocity is the same for both cases, the only factor that will determine which has the most kinetic energy is the moment of inertia. In case 1 all of the mass of the hoop is a fixed distance R from the axis, meaning that all of the mass has the same r^2 multiplier as far as the moment of inertia is concerned. In case 2 the mass of the hoop is AT MOST a distance R from the axis--only two points on the hoop qualify. This means that the moment of inertia of this case MUST be less than the moment of inertia of case 1. C) R does not change in either case, as does mass, so the KE for both systems is the same.

  15. ACT x M dx L A mass Mis uniformly distributed over the length Lof a thin rod. The mass inside a short element dxis given by: A)B)C)D)

  16. ACT x M dx L A mass Mis uniformly distributed over the length Lof a thin rod. The contribution to the rod’s moment of inertia provided by element dxis given by: A)B)C)

  17. ACT A disk has a radius R. The area of a thin ring inside the disk with radiusrand thickness dris: A)B)C) One way: • Other ways: • Taylor Series expansion • Circumference ✕ thickness r dr

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