1. Welcome back to Physics 215
Today’s agenda:
Moment of Inertia
Rotational Dynamics
Angular Momentum

2. Current homework assignment
HW8:
Knight Textbook Ch.12: 8, 26, 54, 58, 60, 62, 64
Due Monday, Nov. 5th in recitation

3. compare this with Newton’s 2nd law
Discussion
Dw/Dt) S miri2 = tnet
a - angular acceleration
Moment of inertia, I
I a = tnet
compare this with Newton’s 2nd law
M a = F

4. Moment of Inertia
* I must be defined with respect to a particular axis

5. Moment of Inertia of Continuous Body
Dm a 0 r Dm

6. Tabulated Results for Moments of Inertia of some rigid, uniform objects
(from p.299 of University Physics, Young & Freedman)

7. Parallel-Axis TheoremD D CM
*Smallest I will always be along axis passing through CM

8. Practical Comments on Calculation of Moment of Inertia for Complex Object
To find I for a complex object, split it into simple geometrical shapes that can be found in Table 9.2
Use Table 9.2 to get ICM for each part about the axis parallel to the axis of rotation and going through the center-of-mass
If needed use parallel-axis theorem to get I for each part about the axis of rotation
Add up moments of inertia of all parts

9. Beam resting on pivot SF = S = Vertical equilibrium?
CM of beam N r r rm x m
M = ?
Mb = 2m
Vertical equilibrium?
SF =
S =
Rotational equilibrium?
M =
N =

10. Suppose M replaced by M/2 ?
SF =
vertical equilibrium?
rotational dynamics?
net torque?
which way rotates?
initial angular acceleration?
S =

11. Moment of Inertia? I = Smiri2 * depends on pivot position! I =
* Hence a=t=

12. Rotational Kinetic Energy
K = Si(1/2mivi2 = (1/2)w2Simiri2
Hence
K = (1/2)I w2
This is the energy that a rigid body possesses by virtue of rotation

13. Spinning a cylinder Use work-KE theorem W = Fd = Kf = (1/2)I 2
Cable wrapped around
cylinder. Pull off with
constant force F. Suppose
unwind a distance d of cable 2R F
What is final angular speed of cylinder?
Use work-KE theorem
W = Fd = Kf = (1/2)I 2
Mom. of inertia of cyl.? -- from table: (1/2)mR2
from table: (1/2)mR2
 = [2Fd/(mR2/2)]1/2 = [4Fd/(mR2)]1/2

14. cylinder+cable problem -- constant acceleration method
extended free body diagram N F
* no torque due to N or FW
* why direction of N ?
* torque due to t= FR
* hence a= FR/[(1/2)MR2]
= 2F/(MR)
 wt = [4Fd/(MR2)] 1/2 FW
radius R
 = (1/2)t2 = d/R; t = [(MR/F)(d/R)]1/2

15. Angular Momentum can define rotational analog of linear
momentum called angular momentum
in absence of external torque it will be conserved in time
True even in situations where Newton’s laws fail ….

16. Definition of Angular Momentum
* Back to slide on rotational dynamics:
miri2Dw/Dt = ti
* Rewrite, using li = miri2w
Dli/Dt = ti
* Summing over all particles in body:
DL/Dt = text
L = angular momentum = I w w ri Fi
pivot mi

17. An ice skater spins about a vertical axis through her body with her arms held out. As she draws her arms in, her angular velocity
1. increases
2. decreases
3. remains the same
4. need more information

18. Angular Momentum 1. Point particle:q r p O
Point particle:
|L| = |r||p|sin(q) = m|r||v| sin(q)
vector form  L = r x p
– direction of L given by right hand rule
(into paper here)
L = mvr if v is at 900 to r for single particle

19. Angular Momentum 2. rigid body: * |L| = I w(fixed axis of rotation)
* direction – along axis – into paper here

20. Rotational Dynamics t=Ia DL/Dt = t These are equivalent statements
If no net external torque: t=0 
* L is constant in time
* Conservation of Angular Momentum
* Internal forces/torques do not contribute
to external torque.

21. Bicycle wheel demo Spin wheel, then step onto platform
Apply force to tilt axle of wheel

22. Linear and rotational motion
Torque
Angular acceleration
Angular momentum
Kinetic energy
Force
Acceleration
Momentum
Kinetic energy

23. A hammer is held horizontally and then released. Which way will it fall?

24. General motion of extended objects
Net force  acceleration of CM
Net torque about CM  angular acceleration (rotation) about CM
Resultant motion is superposition of these two motions
Total kinetic energy K = KCM + Krot

25. Three identical rectangular blocks are at rest on a flat, frictionless table. The same force is exerted on each of the three blocks for a very short time interval. The force is exerted at a different point on each block, as shown.
After the force has stopped acting on each block, which block will spin the fastest?
1. A.
2. B.
3. C.
4. A and C.
Top-view diagram

26. Three identical rectangular blocks are at rest on a flat, frictionless table. The same force is exerted on each of the three blocks for a very short time interval. The force is exerted at a different point on each block, as shown.
After each force has stopped acting, which block’s center of mass will have the greatest speed?
1. A.
2. B.
3. C.
4. A, B, and C have the same C.O.M. speed.
Top-view diagram

27. Reading assignment Rotational dynamics, rolling
Remainder of Ch. 12 in textbook