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

center of mass center of gravity An object naturally spins around its: 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?

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) Equal mass L/R & U/D Actual CM 1 kg 2 kg Geometric center shown at line

Multiply the mass times its distance from the center of mass Actual CM 1 kg 2 kg 1 m 1 m 1 m 2 m * 1 kg 1 m * 2 kg

A massive sun and 3 small planets. Where is the center of mass? Sun CM

The solar system does NOT rotate around the center of the 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

Fg = mg BUT you can PRETEND like the Force of gravity 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 ?

YES Would the object at the right fall over 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

Still a little sinking left before TIMBER!!! The CM is still under the base

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

Where is the book most stable physics physics 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?

Which is more stable This is more stable for 2 reasons #1 a lower CG

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

If the book is more stable here physics physics physics Why does it not fall when upright?

CM animation

Stable equilibrium 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

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) Equal mass L/R & U/D Actual CM Geometric center shown at line

Book questions page 147 9, 12, 14, 17, 18, 19, 23, 24, 26