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Announcements. Quiz 4 on Friday Covers Unit 7 (Work and Energy) and Unit 8 (Potential Energy ). Lecture 10: Center of Mass. Today’s Concept: Center of mass Finding it Using it. Center of Mass. Cube Example. Cube Example: Using Volume Mass Density. ρ = mass density (mass/volume).

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Presentation Transcript
announcements
Announcements
  • Quiz 4 on Friday
    • Covers Unit 7 (Work and Energy) and Unit 8 (Potential Energy)
lecture 10 center of mass

Lecture 10: Center of Mass

Today’s Concept:

Center of mass

Finding it

Using it

cube example using volume mass density
Cube Example: Using Volume Mass Density

ρ= mass density

(mass/volume)

exercise
Exercise

Show mathematically that the center of mass of a uniform stick of length Land mass Mis at x = L/2.

dm

x = L

x = 0

exercise1
Exercise

Show mathematically that the center of mass of a uniform stick of length Land mass Mis at x = L/2.

dm

x = L

x = 0

act disk vs 2 half disks
ACT: Disk vs. 2 half-disks

The disk shown in Case1 clearly has its CMat the center. Suppose the disk is cut in half and the pieces arranged as shown in Case2

In which case is the center of mass highest?

A) Case 1 B) Case 2 C) same

x

xCM

Case1

Case2

act disk vs 2 half disks1
ACT: Disk vs. 2 half-disks

The disk shown in Case1 clearly has its CMat the center. Suppose the disk is cut in half and the pieces arranged as shown in Case2

In which case is the center of mass highest?

A) Case 1 B) Case 2 C) same

x

x

x

xCM

x

Case1

Case2

kinematic quantities
Kinematic Quantities

If the net external force is zero, the CMwon’t accelerate

act system of particles
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?

A) 0 0

B) 0 H/2

C) 0 H/3

D) H/4 H/4

E) H/4 0

y

x y

H

x

slide14
C) 0 H/3

y

X Y

H

(symmetry)

x

homework problem
Homework Problem

= 0.975 m

= -1 m

checkpoint system of two masses
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.

2F

F

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.

checkpoint revote system of two masses
CheckPointRevote: System of Two Masses

2F

F

M

2M

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.

act system of 2 masses
ACT: System of 2 Masses

Two objects, one having twice the mass of the other, are initially at rest. A force Fpushes the mass Mto the right. A force pushes the mass 2M upward. What is the acceleration of the center of mass?

A) B)C)D)

M

2M

F

checkpoint two men pulling
CheckPoint: Two Men Pulling

Two guys who weigh the same are holding onto a masslesspole 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?

act 2 guys pulling ii
ACT: 2 guys pulling, II

M

2M

A large skinny guy with mass 2Mand a smaller guy with mass Mare holding onto a masslesspole 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

-3 m

0 m

3 m

Where is the center of mass of the system initially?

Does the center of mass accelerate?

act 2 guys pulling ii1

M

2M

ACT: 2 guys pulling, II

A large skinny guy with mass 2Mand a smaller guy with mass Mare holding onto a masslesspole 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

-3 m

0 m

3 m

m

m

= -1 m