Download Goal: To understand momentum

Goal: To understand angular
motions
Objectives:
1) To learn about Circular Motions
2) To learn about Rotational Inertia
3) To learn about Torque
4) To examine Center of Mass
5) To learn about what causes Stability
6) To understand the difference between Centripetal
Force vs Centrifugal force
7) To understand Angular Momentum
8) To understand the Conservation of Angular
Momentum
9) To understand the Affects on Earth due to the
conservation of angular momentum
Circular Motion
• Previously we examined speed and
velocity.
• However these were movements in a
straight line.
• Sometimes motions are not straight, but
circular.
Around and around
• If you rotate in a circle there will be a rate you
rotate at.
• That is, you will move some angle every second.
• w = angular velocity = change in angle / time
• Units of w are radians/second or
degrees/second
• If you want a linear speed, the conversion is:
• V = radius * angular velocity (in radians /
second)
Lets do an example.
• You are 0.5 m from the center of a merry-goround.
• If you go around the merry-go-round once every
3.6 seconds (hint, how many degrees in a circle)
then what is your angular velocity in
degrees/second.
• There are 2 pi radians per circle. What is your
angular velocity in radians per second?
• What is your linear velocity in meters per
second?
Rotational Inertia
• If you want to know how something will
accelerate linearly you need to know the
force and mass.
• For circular acceleration the equivalent of
the mass is called Rotational Inertia.
• Newton’s First law also applies here.
• Something in rotation stays there unless
you act upon it.
Equations…
• For a very small in size object traveling in a
circle the inertia for the object is:
• Inertia = mass * radius * radius
• Where radius is the radius of the circle it is
moving in.
• For any not small object the inertia depends on
how much of the mass is far from the point you
are rotating around.
• The more mass further out the higher the inertia
(the harder it is to spin something).
Getting into Inertia Shape
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A solid ball: I = 2/5 m r * r
A solid cylinder: I = ½ m r * r
A meter stick from an end: I = 1/3 m L * L
(L = length of stick)
A meter stick rotating around its center:
I = 1/12 m L * L
• A hoop spinning around its center:
I=mr*r
• A hoop spinning on its side:
I=½mr*r
A ball of mass 1.2 kg
• And a radius of 0.2 m.
• What is its Inertia if it is solid?
• If an ant of mass 0.01 kg is on the edge of
the ball what is the inertia of the ant?
Torque
• Now that we know about rotational mass
we can examine rotational force!
• First of all lets see rotational acceleration:
• Rotational acceleration = change in
rotational velocity / time
• Torque = force * distance from rotation pt
• Torque = Inertia * rotational acceleration
The torque challenge!
• A 30 kg kid sits on one end of a seesaw at
a distance of 2.4 m from the center.
• A bigger kid, 60 kg, thinking for some
reason that if he gets closer to the center
that he can push more weight around get
0.7 m from the center.
• Which kid has more torque?
• Who will end up in the air?
Center of Mass
• Objects that are not tied down or held on
an end will rotate around their center of
mass.
• The center of mass is the average position
of mass for an object.
• Note that for a weirdly shaped object, the
center of mass can actually occur in a
place where there is no mass.
• Where is the center of mass for a hollow
soccer ball?
Sit down, stand up
• If I crouch down. What happens to my
center of mass?
• When is this a good thing?
• When is this a bad thing?
• What happens to my center of mass if I
lean forward?
Stability
• If I lean over at what point will I fall over?
Stability
• If I lean over at what point will I fall over?
• If my center of weight in the horizontal
direction is at a point that is no longer
supported by my base (such as my feet)
then I fall over.
• Why do I fall over?
To support
• You just need to make sure your center of
mass is supported.
Centripetal vs Centrifugal force
• These two are very similar.
• Centripetal force is a force that pulls you to
the center.
• Gravity is an example here.
• When you are in circular motion,
centrifugal force will try to push you out,
and cancels out the centripetal force.
Equation
• Centrifugal force = mass * velocity * velocity /
radius
• A spacecraft is in orbit around the earth at a
distance of 6.5 * 106 m and a velocity of 8.2 *
103 m/s.
• A) If the mass of the spacecraft is 3000 kg then
what is the centrifugal force on the spacecraft?
• B) How does that force compare to the
gravitational force on the spacecraft (yes use mg
here)?
Angular Momentum
• Did you notice that when I was talking
about angular velocity, acceleration, and
force, I left out momentum.
• Well, no longer.
• Just like with the other values,
• Angular momentum = Inertia * Angular
velocity
• (just live normal momentum = mv)
And in case you are wondering…
• Yes, angular kinetic energy = ½ Inertia * angular velocity
* angular velocity
• But back to Momentum:
• Angular mom = Inertia * Angular velocity
• And remember that:
• Angular velocity = velocity / radius
• Inertia = mass * radius * radius
• So, therefore,
• Angular Momentum (L) = mass * velocity * radius
Conservation of angular
momentum
• Just like with normal momentum, angular
momentum is conserved!
• What does this mean?
• Well, if you rotate, you stay rotating with
constant angular momentum.
• If you spin around the earth, you stay
spinning!
Ice skater
• An ice skater puts our their hands and spins.
• Ang mom = Inertia * angular velocity.
• The skater then pulls their hands towards their
body.
• A) What happens to the Inertia (remember that
inertia is greater when mass is further from the
rotational point)?
• B) Knowing that Angular momentum is
conserved what must happen to the angular
velocity?
Diver
• A diver with a height of 2 meters and a mass of
80 kg dives off a diving board.
• Initially he rotates around his center at an
angular velocity of 3 radians/second.
• A) If his moment of inertia = 1/12 m * L * L then
what is his moment of inertia and what is his
angular momentum?
• B) He tucks into a ball with a radius of 0.5 m.
Now the moment of Inertia = 2/5 m * r *r then
what is his moment of Inertia?
• C) Knowing that angular momentum is
conserved, what is his new angular velocity?
Hadley circulation
• http://ess.geology.ufl.edu/ess/Notes/AtmosphericCirculation/atmoscell_big.jpeg
As air moves North or
South, it moves E/W
because of the spin
of the earth.
Going up in
Latitude means
you have less
rotational Energy
(smaller radius).
Therefore, to
conserve energy,
the air moves
westward.
Hurricanes
Conclusion
• Well, we have learned everything we could
possibly want to know about angular
motions.
• We see that once you get the inertia – or
the rotational equivalent to mass, that all
the equations for rotations are the same
as for non rotations.
• Angular momentum is conserved, and this
affects our weather – but no it does NOT
affect our toilets!