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Balanced Rock Arches National Park Moab, Utah

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Chapter 10 Center of Gravity

It doesn’t seem stable why doesn’t it fall?

Balanced Rock

Arches National Park

Moab, Utah

When a ball is spun in the air it spins around its center

& so does a block of wood

That is to say the axis of rotation is through the center of the object

A hammer behaves a little differently

An object naturally spins around its:

center of mass

which is very similar to (we will use these interchangeably)

center of gravity

Center of Gravity or Center of Mass

This is a location of the average position of mass in an object

CM

If an object is symmetrical and

uniform (made of the same material throughout)

the center of mass is just the center

CM

If an object has more mass at one end,

the center of mass if shifted that way

CM

Lead

Plastic

Where is the center of mass?

Where is the center of mass?

Where would the CM of a large pot be?

Equal mass L/R & U/D

Actual CM

But you can’t just say that there is the same amount of mass to the left and to the right

Where do you think this set of masses connected by a string would rotate about if swung in the air

At what point would there be the same mass to the left & right (assume the string is massless)

1 kg

2 kg

Geometric center shown at line

2 m * 1 kg

1 m * 2 kg

Multiply the mass times its distance from the center of mass

Actual CM

1 kg

2 kg

1 m

1 m

1 m

CM

A massive sun and 3 small planets.

Where is the center of mass?

Sun

Sun

The solar system does NOT rotate around the center of the sun.

Sun

The sun actually wobbles a bit because…

Does the center of mass of our solar system change or is it constant?

Sun

Compare the center of mass after the explosion?

If there are no outside forces, the center of mass is the same

What if the fragments are different sizes?

Why does a washing machine vibrate during the spin cycle sometimes but not others?

Where is the center of mass

If a puck is pushed across a frictionless table

It moves in a straight line at a constant speed

A motion graph for this is…

If a wrench is slid across a frictionless table

The wrench below spins about is CM

And its CM moves in a straight line at a constant speed

If a baseball is thrown into the air it follows a parabolic path

If a bat is tossed it looks a little more chaotic but….

But really just 2 things happen

1.) the object rotates around its CM

2.) the CM follows a parabolic path

If ball is thrown into the air, it follows a parabolic path (sorry can’t control the speed any better)

The hammer does 2 things when tossed

1.) rotates around the CM

2.) the CM follows the typical path

Now at the same time.

A hammer thrown into the air

The center of mass / center of gravity

is special for several reasons.

FOR EXAMPLE:

Gravity pulls down on an entire meter stick (each atom)

Fg

BUT this would make calculations a bit tiresome

FOR EXAMPLE:

Fg = mg

BUT you can PRETEND like the Force of gravity

is only pulling down on the CENTER OF MASS

FN = mg

Fg = mg

When a fulcrum is placed under the CM,

It balances!!!

(the net force is zero)

This is an easy way to find the center of mass of an object

JUST BALANCE IT ON YOUR FINGER

If a weight is hung at one end, what happens to the CM

So the object isn’t balanced

In order to be balanced the pivot must be in line with the CM

What if the mass was dropped down by a string?

What would happen to the CM?

Would it still balance?

LET’s Try it

As long as the CM of mass and the pivot point

Are in line with gravity, it will balance

Another interesting fact about the CM, it that if an object is hung by a point with a plum bob attached. The line will ALWAYS run through the CM.

A Gravity Vector drawn from the pivot point will always pass through the CM (when balanced)

Why doesn’t the leaning tower of pisa fall?

Would the object at the right fall over

?

Would the object at the right fall over

YES

Fg

Fg

An object will topple if a vertical line through its CG is not under the support base

This block can’t be tilted anymore without falling

Fg

DEMO- Tilting box

The CM is still under the base

Still a little sinking left before TIMBER!!!

A bus MUST withstand an angle of 28o without tipping

Some can withstand 40o

28o

Center line

40o

The CM must be lower for a greater tipping angle

Highest CM for 28o

Highest CM for 40o

How do the manufacturers get the CM well below the

halfway mark, even with passengers on the top deck?

Highest CM for 40o

Cargo in the back of truck should always be secured

OK

If it isn’t secured or strapped in then…

OK

UH OH

Activity:

Place your feet on the floor in front of you chair.

Now stand up BUT notice what you do before standing!!!

Demos: balanced boards

hammer paint stick

physics

physics

Where is the book most stable

A system is most stable with the CM as low as possible.

There is a tendency towards this state.

Demo: rolling concrete

If a ping pong ball is pushed into the water in a glass what happens to the CM?

This is more stable for 2 reasons

#1 a lower CG

Which is more stable

The object at right must have its CM go UP before falling

The object at left does not

Unstable equilibrium

Stable equilibrium

Any movement lowers the CM

The CM must rise first

physics

physics

physics

If the book is more stable here

Why does it not fall when upright?

CM animation

The object is forced to pivot at the fulcrum.

Any rotation causes the CM to rise

Stable equilibrium

What about like this?

UNstable equilibrium

What about here?

This object is in neutral equilibrium..

The CM doesn’t rise or fall as it rotates

Equal mass L/R & U/D

Actual CM

But you can’t just say that there is the same amount of mass to the left and to the right

Where do you think this set of masses connected by a string would rotate about if swung in the air

At what point would there be the same mass to the left & right (assume the string is massless)

Geometric center shown at line

Book questions

page 147

9, 12, 14, 17, 18, 19, 23, 24, 26