1. Congratulations!
You’re a Survivor! We’re Half Way through. Hooray! And We’ve accomplished a LOT already!

2. Today’s Concepts: a) Rotational Motion b) Moment of Inertia
Physics 211 Lecture 14
Today’s Concepts:
a) Rotational Motion
b) Moment of Inertia

4. Summary of Rotations Angular velocity w is measured in radians/sec
Frequency f is measured in revolutions/sec
1 revolution = 2p radians
Period T = 1/f

5. Centripetal vs Angular Accelerationw
Constant a does not mean constant w
Demo: phasor wheel

6. ACT A) T = 2 sec B) T = 2p sec C) T = ½ sec
A disk spins at 2 revolutions/sec.
What is its period?
A) T = 2 sec
B) T = 2p sec
C) T = ½ sec

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

8. CheckPoint
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) B) C) 16 a 8

9. CheckPoint Response
After 4 seconds it has made 4 complete revolutions.
How many revolutions has it made after 8 seconds?
A) B) C) 16 a
A) Because it has constant acceleration, it is making 1 revolution per second. After 8 seconds, it makes 8 revolutions.
B) It is accelerating, making 8 more rotations in the next 4 seconds. so the total is 4+8=12
C) theta = .5 * alpha * t^2 so doubling time increases theta 4 times 9

10. CheckPoint Response
After 4 seconds it has made 4 complete revolutions.
How many revolutions has it made after 8 seconds?
A) B) C) 16 a
What is its angular acceleration, alpha?
Angle theta = (1/2) (alpha)( t2) Don’t forget that time is squared.
4 rev = (1/2) alpha (4 sec)2 = (alpha)(8 sec2)
alpha = (1/2) rev/ sec2
Now, for t = 8 sec; Angle theta = (1/2)((1/2) rev/sec2)(8 sec)2 = 16 rev 10

11. Your Responses good HOW’d you get pi? good But it is accelerating.
How simple?
Good observation.
Tell me a little more.
good
Is it any clearer now?
I don’t understand how pi got in there at all.
But it is accelerating.
Good observation.
Is it any clearer now?
? ? ?
That’s always a good thing to do. Now tell me what you think.
? ? ? 11

12. Calculating Moment of Inertia
Demo: inertia rods
Depends on rotation axis
Demos: Rotating stool &
weights,
Book rubber banded together
What is moment of inertia good for?

13. CheckPoint
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?
A) Ia > Ib > Ic
B) Ia > Ic > Ib
C) Ib > Ia > Ic a b c
Let’s try this again…

14. CheckPoint Which of the following orderings is correct?a
A) Ia > Ib > Ic
B) Ia > Ic > Ib
C) Ib > Ia > Ic b c
Ia = 8mr^2 Ib = 3mr^2 Ic = 4mr^2
Could use parallel-axis theorem (next lecture).
CM is 1/3 up from c. b is closest, then c, then a.

15. Your Responses good Also good Any better by now? ? ? ? ? ? ? ? ?
No, it is distance-squared, r^2 or r*r .
Largest “moment of intertia”, not “solid”.
Yes, but that is the point of the question.
What about location?
good
good

16. Calculation Moment of Inertia
The idea that a spherical shell has a higher moment of inertia doesn't really make sense to me because it has less mass than a solid sphere.
All these depend upon the mass M. “Bigger” is when the mass is farther out; keep the mass M the same.

17. ACT
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 B) Case C) Same w R
Case 2
Case 1
Only about half got this right so let’s try again… 17

18. w In which case does the spinning hoop have the most kinetic energy?
A) Case B) Case C) Same w R
Case 2
Case 1
A) all of the mass is concentrated at the farthest point away from the axis in case 1 C) Since they have the same mass and same angular velocity their kinetic energy would be the same.
Demo: hula hoop 18

19. w In which case does the spinning hoop have the most kinetic energy?
A) Case B) Case C) Same w R
Case 2
Case 1
But the moments of inertia are not the same.
Demo: hula hoop 19

20. But the linear velocity of the pieces is not same. Not same
good
But the linear velocity of the pieces is not same.
Not same
How? What? Why? This doesn’t tell me anything.
Uh-oh; the moments of inertia are quite different.
Uh-oh; the moments of inertia are quite different.
Is it any clearer by now?
Moments of inertiav are not the same.
Uh-oh; the moments of inertia are quite different.
Moments of inertiav are not the same.
good
Is it any clearer by now?
Is it any clearer by now?
Tell me more.
Tell me more. 20

21. ACT
A mass M is uniformly distributed over the length L of a thin rod. The mass inside a short element dx is given by:
A) B) C) D) dx x L M
Demo: meterstick

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

23. ACT
A disk has a radius R. The area of a thin ring inside the disk with radius r and thickness dr is:
A) B) C) r dr

24. (i)
(ii)
(iii)
Using (ii)
Using (i)

25. (iv)
(v)
Use (iv)
Use (v) 25

26. (vi)
(vii)
(viii)
(ix)
Use (viii)
Use (ix) 26

27. (vi)
(vii)
(viii)
(ix)
Use (vii)
Use (vi) 27