1. Announcements Quiz 4 on Friday
Covers Unit 7 (Work and Energy) and Unit 8 (Potential Energy)

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

3. Center of Mass

4. Cube Example

5. Cube Example: Using Volume Mass Density
(mass/volume)

6. Cube Example: CM Calculation

7. Exercise
Show mathematically that the center of mass of a uniform stick of length L and mass M is at x = L/2. dm
x = 0
x = L

8. Exercise
Show mathematically that the center of mass of a uniform stick of length L and mass M is at x = L/2. dm
x = 0
x = L

9. Center of Mass for System of Objects

10. ACT: Disk vs. 2 half-disks
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

11. ACT: Disk vs. 2 half-disks
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

12. Kinematic Quantities
If the net external force is zero, the CM won’t accelerate

13. ACT: System of particles
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? x y H
x y
A) 0 0 B) 0 H/2 C) 0 H/3 D) H/4 H/4 E) H/4 0

14. y
X Y
C) H/3 H
(symmetry) x

15. Homework Problem
= m
= -1 m

16. Homework Problem

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

18. A B C m5
CM moves toward m5 cm

19. CheckPoint: System of Two Masses
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. F 2F M 2M
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
About 42% answered correctly.

20. CheckPoint Revote: System of Two Masses
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) The actual masses are accelerating away from each other at the same rate so the center of mass does not change.
D) CM acceleration is defined as the sum of the total forces divided by the total mass. Total mass is 3m. The forces are not equal, therefore the greater force will determine the direction of acceleration.
E) The acceleration of the center of mass is equal to the external force divided by the total mass.

21. ACT: System of 2 Masses A) B) C) D) F
Two objects, one having twice the mass of the other, are initially at rest. A force F pushes the mass M to the right. A force pushes the mass 2M upward. What is the acceleration of the center of mass? M 2M F
A) B) C) D)

22. CheckPoint: Two Men Pulling
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
Where is the center of mass of the system initially?
Does the center of mass accelerate?

23. ACT: 2 guys pulling, II 2M M A) -3 m B) -1 m C) 0 m D) 1 m E) 3 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
Where is the center of mass of the system initially?
Does the center of mass accelerate?

24. ACT: 2 guys pulling, II m = -1 m A) -3 m B) -1 m C) 0 m D) 1 m E) 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?
A) -3 m
B) -1 m
C) 0 m
D) 1 m
E) 3 m 2M M
-3 m
0 m
3 m m
= -1 m