1. Hour Exam 1 / THIS Thursday
1:30-2:20 pm
In this room, 141 Loomis
Covers Lectures 1-7 only
Bring your calculator!
No “Potential Energy, Conservation or CM”
on this exam
Hour Exam 1 / THIS Thursday
Exam review in class Wednesday – we will work through problems from practice exams (student choice). You bring the questions!

2. Today’s Concept: Center of Mass Finding it Using it
Physics 211 Lecture 10
Today’s Concept:
Center of Mass
Finding it
Using it

3. Not just for “point” particles, but for ANYTHING!
Center of MAss
Not just for “point” particles,
but for ANYTHING!
(if you use center of mass)
Demos: Center of mass rod w/ spheres
Rubber pork chop

4. Center of Mass

5. Center of Mass Discrete Particles

6. ACT XCM YCM y A) 0 0 B) 0 H/2 C) 0 H/3 D) H/4 H/4 E) H/4 0 x
Three tiny equal-mass magnets are placed on a horizontal frictionless surface at the corners of an equilateral triangle. When the magnets are released, they attract and quickly slide to a single point. What are the coordinates of that point?
I would like more examples of center of mass calculations.
XCM YCM y A)
B) H/2
C) H/3
D) H/4 H/4
E) H/ H x

7. Three tiny equal-mass magnets are placed on a horizontal frictionless surface at the corners of an equilateral triangle. When the magnets are released, they attract and quickly slide to a single point. What are the coordinates of that point? y
XCM YCM
C) H/3 H
(symmetry) x

8. Homework Problem
New
= m
= -1 m

9. Homework Problem

10. A B C D m5
CM moves toward m5 cm

11. A B C m5
CM moves toward m5 cm

12. Center of Mass (Continuous)
Mass Density

13. Example Uniform Stick
A uniform stick of length L and mass M.
x=0
x=L

14. CheckPoint
A yummy glazed doughnut is shown below. Where is the center of mass of this fantastic culinary delight?
A) In the center of the hole.
B) Anywhere along the blue dashed line going through the solid part of the dough.
C) The center of mass is not defined in cases where there is missing mass. x
The yummy doughnut is round and has equal weight distribution all over the doughnut, therefore, the center of mass must be in the middle

15. Your Comments
But there’s always a C of M.

16. Cube Example
HO 05
The triple integral was a perfect recipe for freshmen destruction.
ARgghgrrggggh.....triple.....int-int-int-integral!? :: ERROR ERROR ERROR :: :: BREAK PROCESS :: :: Rebooting neural net... :: ::

17. Cube Example: Using Volume Mass Density

18. Cube Example: CM Calculation

19. Center of Mass for System of Objects
I would like to discuss finding the CM of objects that aren't nice, easy shapes like boxes and spheres are. Something cool, like sharks with laser beams attached to their heads.

20. ACT
The disk shown in Case 1 clearly has its CM at the center. Suppose the disk is cut in half and the pieces arranged as shown in Case 2
In which case is the center of mass highest?
A) Case B) Case C) same x
x CM
Case 1
Case 2

21. ACT
The disk shown in Case 1 clearly has its CM at the center. Suppose the disk is cut in half and the pieces arranged as shown in Case 2
In which case is the center of mass highest?
A) Case B) Case C) same x x x
x CM x
Case 1
Case 2

22. F=MA for Center of Mass!
If the total external force is zero, the CM won’t accelerate.

23. Center of Gravity
For object small enough that g is constant, the center of mass is the same as the center of gravity. The gravity force acting on all the parts of the object acts the same as if the entire mass were at the center of mass.
Demos: yellow c.m. shapes
Hammer on hinged board
Leaning tower

24. CheckPoint
Two objects, one having twice the mass of the other, are initially at rest. Two forces, one twice as big as the other, act on the objects in opposite directions as shown. 2F 2M M F
Which of the following statements about the acceleration of the center of mass of the system is true:
A) aCM = F/M to the right
B) aCM = F/(3M) to the right
C) aCM = 0
D) aCM = F/(3M) to the left
E) aCM = F/M to the left

25. CheckPoint 2F 2M M F
Which of the following statements about the acceleration of the center of mass of the system is true:
C) aCM = 0
D) aCM = F/(3M) to the left
E) aCM = F/M to the left
C) F=Ma. The 2M and the 2F would cancel each other out, making the bigger object accelerate at the same rate as the smaller ball. Therefore, the acceleration would be zero.
D) Acceleration is equal to the net force divided by the total mass. The net force is F to the left and the total mass is 3M
E) ACM = F total/M total = 3F/3M = F/M

26. Your Comments ! ! ! ! ! Good, clear, fine.
The two forces are NOT equal; they are F and 2F
? ? ? ? ?
No; but your comment confuses me.

27. ACT
Two pucks of equal mass, on a frictionless table, are being pulled at different points with equal forces. Which one gets to the end of the table first?
A) Puck 1 B) Puck 2 C) Same 1)
F = 2T M T 2)

28. ACT
Two pucks of equal mass, on a frictionless table, are being pulled at different points with equal forces. Which one gets to the end of the table first? a1
C) Same 1)
F = 2T M T a2 2)
SAME

29. CheckPoint
HO 16
Two guys who weigh the same are holding onto a massless pole while standing on horizontal frictionless ice. If the guy on the left starts to pull on the pole, where do they meet?
-3 m
0 m
3 m
A) -3 m B) 0 m C) 3 m

30. Your Comments
What IS a linear system?
! ! !
! ! !
? ? ?
? ? ?
? ? ?

31. ACT 2M M A) -3 m B) -1 m C) 0 m D) 1 m E) 3 m -3 m 0 m 3 m
A large guy with mass 2M and a smaller guy with mass M are holding onto a massless pole while standing on frictionless ice, as shown below. If the little guy pulls himself toward the big guy, where would they meet? 2M M
A) -3 m
B) -1 m
C) 0 m
D) 1 m
E) 3 m
-3 m
0 m
3 m

32. m m = -1 m 2M M A) -3 m B) -1 m C) 0 m D) 1 m E) 3 m -3 m 0 m 3 m
A large skinny guy with mass 2M and a smaller guy with mass M are holding onto a massless pole while standing on frictionless ice, as shown below. If the little guy pulls himself toward the big guy, where would they meet? 2M M
A) -3 m
B) -1 m
C) 0 m
D) 1 m
E) 3 m
-3 m
0 m
3 m m m
= -1 m

33. Main Points