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Mechanics Lecture 10, Slide 1

Announcements Quiz 4 on Friday

Covers Unit 7 (Work and Energy) and Unit 8 (Potential Energy)

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Mechanics Lecture 10, Slide 2

Lecture 10: Center of Mass

Today’s Concept:Center of mass Finding it Using it

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Mechanics Lecture 10, Slide 3

Center of Mass

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Mechanics Lecture 10, Slide 4

Cube Example

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Mechanics Lecture 10, Slide 5

Cube Example: Using Volume Mass Density

ρ = mass density(mass/volume)

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Mechanics Lecture 10, Slide 6

Cube Example: CM Calculation

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Mechanics Lecture 10, Slide 7

x = Lx = 0

ExerciseShow mathematically that the center of mass of a uniform stick of length L and mass M is at x = L/2.

dm

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Mechanics Lecture 10, Slide 8

x = Lx = 0

ExerciseShow mathematically that the center of mass of a uniform stick of length L and mass M is at x = L/2.

dm

L

dxMdm dx

L

M

xdmM

xcm

1

Ldx

L

Mx

M 0

1 L

xdxL 0

1

L

xL 0

2

2

11

22 0

2

11 L

L 2

L

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Mechanics Lecture 10, Slide 9

Center of Mass for System of Objects

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Mechanics Lecture 10, Slide 10

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?

ACT: Disk vs. 2 half-disks

Case 1 Case 2

xCM

x

A) Case 1 B) Case 2 C) same

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Mechanics Lecture 10, Slide 11

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 1 B) Case 2 C) same

ACT: Disk vs. 2 half-disks

Case 1 Case 2

xCM

x

x

x

x

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Mechanics Lecture 10, Slide 12

If the net external force is zero, the CM won’t accelerate

Kinematic Quantities

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Mechanics Lecture 10, Slide 13

ACT: System of particles

A) 00

B) 0H/2

C) 0H/3

D) H/4H/4

E) H/40

x y

x

y

H

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?

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Mechanics Lecture 10, Slide 14

C) 0 H/3

X Y

(symmetry)

x

y

H

0CMX

3

H

321

332211

mmm

ymymymYCM

m

Hmmm

3

00

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Mechanics Lecture 10, Slide 15

= -1 m

Homework Problem

= 0.975 m 4321

44332211

mmmm

xmxmxmxmxcm

4321

44332211

mmmm

ymymymymycm

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Mechanics Lecture 10, Slide 16

Homework Problem

4321

,44,33,22,11, mmmm

vmvmvmvmv xxxx

xcm

4321

,44,33,22,11, mmmm

vmvmvmvmv yyyy

ycm

2,

2, ycmxcmcm vvv

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Mechanics Lecture 10, Slide 17

m5 CM moves toward m5

ABCD

cm

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Mechanics Lecture 10, Slide 18

m5 CM moves toward m5

cm

ABC

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Mechanics Lecture 10, Slide 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.

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

Which of the following statements about the acceleration of the center of mass of the system is true?

F2F M2M

About 42% answered correctly.

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Mechanics Lecture 10, Slide 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) 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.

C) ACM = 0

D) ACM = F/(3M) to the left

E) ACM = F/M to the left

F2F M2M

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Mechanics Lecture 10, Slide 21

ACT: System of 2 Masses

A) B) C) D)

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?

M2MF

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Mechanics Lecture 10, Slide 22

CheckPoint: Two Men Pulling

A) -3 m B) 0 m C) 3 m

-3 m 0 m 3 m

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?

Where is the center of mass of the system initially?

Does the center of mass accelerate?

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Mechanics Lecture 10, Slide 23

ACT: 2 guys pulling, II

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?

-3 m 0 m 3 m

2M M

Where is the center of mass of the system initially?

Does the center of mass accelerate?

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Mechanics Lecture 10, Slide 24

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?

-3 m 0 m 3 m

2M M

M

MM

3

)3()3(2 m m = -1

m21

2211

mm

xmxmXCM

ACT: 2 guys pulling, II