Levers, Torque, Rotation, Angular momentum,

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Levers, Torque, Rotation, Angular momentum,
Lever and mechanical advantage Torque Rotating
stuff Angular momentum
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Remember Conservation of Energy
Controlled stored energy Potential Energy
Can be converted to Motion energy Kinetic
Energy
Uncontrolled stored energy heat, etc Easy to
make (friction, etc) difficult to recover (ie to
control) but stay tuned.
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Remember Conservation of Momentum

• Dont try this at home
• Car wanted to go straight
• Even significant force from tire traction could
  not enforce Dp
• So slow down BEFORE the curve

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Rotation and center of mass

• Push on an extended object and in general it will
  turn.
• Throw an object and it will rotate and translate

Center of mass is balance point (aka center of
gravity) Use symmetry to locate or
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How do Baryshnikov and MJ do that?!?

• Also skaters, high jumpers,
• leaping predators, etc
• Wave limbs to fly
• or to defy gravity

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Where is the center of mass

• Where is the CM for a book?
• Where is the CM of a football?
• Where is the CM of a DVD?
• Where is the CM of a burrito?
• Where is the CM of a bar-bell?
• Where is the CM of a can of clam chowder?

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Balancing a see-saw

• Older siblings sit closer to the fulcrum
• (or younger siblings wail)

Torque (lever arm) ? (force)
t r ? F?
So the toddler can lift the babysitter with a
long enough lever arm
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Lever Torque around a fulcrum

• Mechanical advantage amplifies the force
• r1F1 r2F2

F2 (r1/r2) ? F1 F2 can be as big as you
like, if you can find a longer stick levers
(like ramps) are old and ubiquitous
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Levers, levers and levers
They are everywhere!

• Different efficiencies
• (or amounts of leverage)
• depending on relations
• of fulcrum and forces

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Levers closer to home

• But wait a minute!
• This gives a mechanical DISadvantage?

Why would nature/evolution choose this method??
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Gear ratios

• Change gears means change ratios of cog
  diameters
• r1T r2T (oops?!? Whats missing here?)
• And hence the rotation rate of the wheel compared
  to the pedal

r1
r2
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Torques in balance

• Only the force component perpendicular to r
• (F? ) contributes to the torque

F
r
Net torque 0
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Torque equation

• What is the objects response to the torque?
• It starts to rotate, or changes the rate of
  rotation
• In analogy with Newton II
• (and in agreement with experiments!)

t Ia
Instead of mass there is a moment of inertia I I
has units (kg)(m2) (mass)(lever arm)2 aka
rotational mass measure of resistance to torque
(like mass is resistance to force) a is the
angular acceleration
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What is the rotational mass?

• Distribution of objects mass (m) around the axis
  of rotation
• I Cmr2

C never exceeds 1 usually between 0.05 and 1
C for a frisbee is bigger than C for a football
(but m is smaller) Anyway Ill give you the
number
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Vector rotation?!?!?

• Why is rotation a vector???
• if it quacks like a duck

Well the axis points in two directions arbitraril
y choose the right-handed direction
Torque (t) and angular acceleration (a) and
angular momentum L Iw are all right-handed
vectors
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Angular Momentum

• Inertia all over again
• tendency to maintain state of rotational motion
• (cf spinning bike wheel, spiraling football,
  frisbee, etc)

As with Newton II t Dt IDw DL
If t 0 then L constant Angular momentum
is conserved
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Conservation of Angular Momentum in action

• Helicopter blades
• Auto engine flywheels
• Potters wheel
• Satellite orbit
• Bicycle wheels

Etc.
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How does a diver start spinning?

• Only place where torque can occur
• when she springs from board

Once she leaves the board, she cant stop or
start spinning! L I w constant She changes w
by changing I.
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Before he pukes

• Change w by changing I

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Used by humans and cleverer animals

• Navigational gyro axis stays fixed
• so how does the cat break the rules??

• The cat pushes it's hind legs out and stretches
  them as far behind itself as possible, causing
  the rear section of the body to have a larger I
  than the front section.
• At the same time it bring in its front legs
  making them have lower I. The front half can
  rotate in one direction, while the rear part
  rotates much less in the opposite direction.
• Once the front section of the cat has rotated the
  180, the cat brings back in the rear legs and
  stretches out the front legs so that I is larger
  in the front, allowing the cat to rotate the rear
  half of its body around 180.
• During the whole maneuver L 0.

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New Newton, New Principles
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Take home messages

• Center of mass center of gravity balance
  point for an object (Symmetry!)
• Torque is t r ? F?. (Only perpendicular
  component matters)
• Torque equation t Ia
• Rotational mass I (mass)(lever arm)2
• Conservation of angular momentum If t 0, then
  L constant
• Right hand rule for rotation vectors