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Lesson #65Lesson #65Topic: Center of Mass / GravityTopic: Center of Mass / Gravity

Objectives: Objectives: (After this class I will be able to)(After this class I will be able to)

1. Define center of gravity / mass1. Define center of gravity / mass

2. Locate an object’s center of mass 2. Locate an object’s center of mass

4/18/07

Warm Up: Toss different objects through the air and observe differences and similarities of their motion. Assignment: Book Problems p147 RQ #5, 7, 8, 11,12

Center of gravity / massCenter of gravity / mass Any object can be treated as a particle with all Any object can be treated as a particle with all

of its mass concentrated at a single point.of its mass concentrated at a single point. This point is called the objects This point is called the objects Center of MassCenter of Mass.. The point where the force of gravity (weight) The point where the force of gravity (weight)

acts on an object is called its acts on an object is called its Center of Center of Gravity.Gravity.

For the purpose of this course, center of gravity For the purpose of this course, center of gravity and the center of mass will be the same thing. and the center of mass will be the same thing.

However, for very large objects these points However, for very large objects these points may be in slightly different places. may be in slightly different places.

Example: the Empire State building has a center Example: the Empire State building has a center of mass that is 1mm higher than its center of of mass that is 1mm higher than its center of gravity.gravity.

Center of MassCenter of Mass

Center of MassCenter of Mass All projectiles will follow a parabolic motion All projectiles will follow a parabolic motion

through the sky. through the sky. However, even if an object explodes in the However, even if an object explodes in the

air, its center of mass will continue to air, its center of mass will continue to follow this motion.follow this motion.

Example: A firework is launched into the Example: A firework is launched into the air and explodes at its highest point.air and explodes at its highest point.

Center of MassCenter of Mass The center of mass of an object can The center of mass of an object can

be located many different ways.be located many different ways. Any time you flip an object of any Any time you flip an object of any

shape or size, it rotates around its shape or size, it rotates around its center of mass.center of mass.

When you balance an object on a When you balance an object on a point, this point must be below the point, this point must be below the object’s center of mass.object’s center of mass.

Lesson #66Lesson #66Topic: Lab: Finding C of GTopic: Lab: Finding C of G

Objectives: Objectives: (After this class I will be able to)(After this class I will be able to)

1.1. Devise a way to find the exact Devise a way to find the exact location of an object’s center of mass.location of an object’s center of mass.

4/19/07

Warm Up: How would you estimate the location of the center of mass of an object?

Lab task: Trace 5 objects on separate Trace 5 objects on separate sheets of paper and mark where the sheets of paper and mark where the center of mass is for each object. center of mass is for each object.

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Lesson #67Lesson #67Topic: Solving for CMTopic: Solving for CM

Objectives: Objectives: (After this class I will be able to)(After this class I will be able to)

1.1. Calculate and solve for the exact Calculate and solve for the exact location of the center of mass of an location of the center of mass of an object.object.

4/20/07

Warm Up: Where is the approximate correct location of the center of mass of a car?Assignment: “Solving CG” due at the end of the period tomorrow.

Where is the approximate correct location of the center of mass of a car?

1.1. AA

2.2. BB

3.3. CC

4.4. DD

A B C D

25% 25%25%25%

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Solving for CM / CGSolving for CM / CG The location of the center of mass of The location of the center of mass of

a system can be found using the a system can be found using the conservation of momentum.conservation of momentum.

Consider a block that explodes into a Consider a block that explodes into a bunch of pieces.bunch of pieces.

M = mM = m1 1 + m+ m2 2 + m+ m3 3 + m+ m44

m1 m2 m3 m4

M

Solving for CM / CGSolving for CM / CG The momentum of the system before The momentum of the system before

exploding is the same after exploding.exploding is the same after exploding. ppbeforebefore=p=pafterafter

MvMvcmcm=m=m11vv11 + m + m22vv22 + m + m33vv33 + m + m44vv44

Remember v=d/t or x/t where x=distanceRemember v=d/t or x/t where x=distance M(xM(xcmcm/t) = m/t) = m11(x(x11/t) + m/t) + m22(x(x22/t) + m/t) + m33(x(x33/t) + /t) +

mm44(x(x44/t)/t) Time is the same for the whole system.Time is the same for the whole system.

nncm xmxmxmMX ...2211

ExampleExample Three masses, 2kg, 3kg, and 6kg, are Three masses, 2kg, 3kg, and 6kg, are

located at positions (3,0), (6,0), and (-4,0) located at positions (3,0), (6,0), and (-4,0) respectively, in meters from the origin. respectively, in meters from the origin. Where is the center of mass of the Where is the center of mass of the system?system?

Three people of roughly equivalent mass m on a Three people of roughly equivalent mass m on a lightweight banana boat sit along the x axis at lightweight banana boat sit along the x axis at positions xpositions x11=1m, x=1m, x22=5m, and x=5m, and x33=6m. Find the =6m. Find the

position of the CM.position of the CM. 25% 25%25%25%1.1. 2m2m

2.2. 3m3m

3.3. 4m4m

4.4. 5m5m

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The centers of a 4kg sphere and a 7.5kg The centers of a 4kg sphere and a 7.5kg sphere are separated by a distance of 1.5m. sphere are separated by a distance of 1.5m. Where is the center of mass of the system?Where is the center of mass of the system?

.75m

from

the

4kg s

p...

.86m

from

the

4kg s

p...

.98m

from

the

4kg s

p...

1.0

2m fr

om th

e 4k

g s...

25% 25%25%25%1.1. .75m from the .75m from the

4kg sphere4kg sphere

2.2. .86m from the .86m from the 4kg sphere4kg sphere

3.3. .98m from the .98m from the 4kg sphere4kg sphere

4.4. 1.02m from the 1.02m from the 4kg sphere4kg sphere

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Two skaters with masses of 65kg and 45kg, respectively, Two skaters with masses of 65kg and 45kg, respectively, stand 8m apart, each holding an end of a rope. The light stand 8m apart, each holding an end of a rope. The light

skater pulls on the rope until the two skaters meet. How far skater pulls on the rope until the two skaters meet. How far does the light skater travel?does the light skater travel?

8m

6.5

m

4.7

3m

3.2

7m

25% 25%25%25%1.1. 8m8m

2.2. 6.5m6.5m

3.3. 4.73m4.73m

4.4. 3.27m3.27m

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2D Example2D Example Find the location of the CMFind the location of the CM mm11=1kg, m=1kg, m22=2kg, m=2kg, m33=3kg, m=3kg, m44=4kg=4kg

(0,0) (4,0)

(0,4)(4,4)

x

y

m1

m2 m3

m4

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Try to put your head against the wall Try to put your head against the wall and stand up with the chair in the and stand up with the chair in the front of the room. front of the room.

Then try to stand with your heels Then try to stand with your heels against the wall and lean over and against the wall and lean over and touch your toes.touch your toes.

Why can’t you do either of these Why can’t you do either of these things?things?

Lesson #68Lesson #68Topic: StabilityTopic: Stability

Objectives: Objectives: (After this class I will be able to)(After this class I will be able to)

1.1. Define stability.Define stability.

2.2. Describe an example of an object that Describe an example of an object that is stable, unstable, and neutral.is stable, unstable, and neutral.

4/20/07

Warm Up: What is an example of an object that is stable, unstable, and neutral?Assignment: Concept Development 10 - 1

Book Problems p147 RQ #16, 17, 20, TE 3, 5

StabilityStability Stable: An object that is Stable: An object that is stablestable is is

one where any rotation in any one where any rotation in any direction will direction will raiseraise the object’s center the object’s center of gravity.of gravity.

StabilityStability Unstable: An object that is Unstable: An object that is unstableunstable

is one where any rotation in any is one where any rotation in any direction will direction will lowerlower the object’s the object’s center of gravity.center of gravity.

StabilityStability Neutral: An object that is Neutral: An object that is neutralneutral is is

one where any rotation in any one where any rotation in any direction will direction will not changenot change the object’s the object’s center of gravity.center of gravity.

Which of the following objects is Which of the following objects is stable?stable?

A B C

33% 33%33%

1.1. AA

2.2. BB

3.3. CC

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Which of the following objects is Which of the following objects is unstable?unstable?

A B C

33% 33%33%

1.1. AA

2.2. BB

3.3. CC

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Which of the following objects is Which of the following objects is neutral?neutral?

A B C

33% 33%33%

1.1. AA

2.2. BB

3.3. CC

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