Rotational Dynamics

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Lecture 16
Rotational Dynamics
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Angular Momentum
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Angular Momentum
Consider a particle moving in a circle of radius
r,
I mr2
L I? mr2? rm(r?) rmvt rpt
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Angular Momentum
For more general motion (not necessarily
circular),
The tangential component of the momentum, times
the distance
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Angular Momentum
For an object of constant moment of inertia,
consider the rate of change of angular momentum
analogous to 2nd Law for Linear Motion
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Conservation of Angular Momentum
If the net external torque on a system is zero,
the angular momentum is conserved.
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Figure Skater

• A figure skater spins with her arms extended.
  When she pulls in her arms, she reduces her
  rotational inertiaand spins faster so that her
  angular momentum is conserved. Comparedto her
  initial rotational kinetic energy, her rotational
  kinetic energy after she pulls in her arms must
  be

a) the same b) larger because shes rotating
faster c) smaller because her rotational inertia
is smaller
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Figure Skater

• A figure skater spins with her arms extended.
  When she pulls in her arms, she reduces her
  rotational inertiaand spins faster so that her
  angular momentum is conserved. Comparedto her
  initial rotational kinetic energy, her rotational
  kinetic energy after she pulls in her arms must
  be

a) the same b) larger because shes rotating
faster c) smaller because her rotational inertia
is smaller
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As her hands come in, the velocity of her arms is
not only tangential... but also radial.
So the arms are accelerated inward, and the force
required times the ?r does the work to raise the
kinetic energy
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Conservation of Angular Momentum
Angular momentum is also conserved in rotational
collisions
larger I, same total angular momentum, smaller
angular velocity
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Rotational Work
A torque acting through an angular displacement
does work, just as a force acting through a
distance does.
Consider a tangential force on a mass in circular
motion
t r F
Work is force times the distance on the arc
s r ??
W s F
W (r ??) F rF ?? t ??
The work-energy theorem applies as usual.
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Rotational Work and Power
Power is the rate at which work is done, for
rotational motion as well as for translational
motion.
Again, note the analogy to the linear form (for
constant force along motion)
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Dumbbell II
a) case (a) b) case (b) c) no difference d)
it depends on the rotational inertia of the
dumbbell

• A force is applied to a dumbbell for a certain
  period of time, first as in (a) and then as in
  (b). In which case does the dumbbell acquire the
  greater energy ?

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Dumbbell II
a) case (a) b) case (b) c) no difference d)
it depends on the rotational inertia of the
dumbbell

• A force is applied to a dumbbell for a certain
  period of time, first as in (a) and then as in
  (b). In which case does the dumbbell acquire the
  greater energy ?

If the CM velocities are the same, the
translational kinetic energies must be the same.
Because dumbbell (b) is also rotating, it has
rotational kinetic energy in addition.
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A 2.85-kg bucket is attached to a disk-shaped
pulley of radius 0.121 m and mass 0.742 kg. If
the bucket is allowed to fall, (a) what is its
linear acceleration? (b) What is the angular
acceleration of the pulley? (c) How far does the
bucket drop in 1.50 s?
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A 2.85-kg bucket is attached to a disk-shaped
pulley of radius 0.121 m and mass 0.742 kg. If
the bucket is allowed to fall, (a) What is its
linear acceleration? (b) What is the angular
acceleration of the pulley? (c) How far does the
bucket drop in 1.50 s?
(a)
Pulley spins as bucket falls
(b)
(c)
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The Vector Nature of Rotational Motion
The direction of the angular velocity vector is
along the axis of rotation. A right-hand rule
gives the sign.
Right-hand Rule your fingers should follow the
velocity vector around the circle
Optional material Section 11.9
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The Torque Vector
Similarly, the right-hand rule gives the
direction of the torque vector, which also lies
along the assumed axis or rotation
Right-hand Rule point your RtHand fingers along
the force, then follow it around. Thumb points
in direction of torque.
Optional material Section 11.9
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The linear momentum of components related to the
vector angular momentum of the system
Optional material Section 11.9
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Applied tangential force related to the torque
vector
Optional material Section 11.9
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Applied torque over time related to change in the
vector angular momentum.
Optional material Section 11.9
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Spinning Bicycle Wheel
You are holding a spinning bicycle wheel while
standing on a stationary turntable. If you
suddenly flip the wheel over so that it is
spinning in the opposite direction, the turntable
will
a) remain stationary b) start to spin in the
same direction as before flipping c) start to
spin in the same direction as after flipping
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What is the torque (from gravity) around the
supporting point? Which direction does it point?
Without the spinning wheel does this make sense?
Why does the wheel not fall? Does this violate
Newtons 2nd?
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Gravity
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Newtons Law of Universal Gravitation
Newtons insight The force accelerating an
apple downward is the same force that keeps the
Moon in its orbit.
Universal Gravitation
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The gravitational force is always attractive, and
points along the line connecting the two masses
The two forces shown are an action-reaction pair.
G is a very small number this means that the
force of gravity is negligible unless there is a
very large mass involved (such as the Earth).
If an object is being acted upon by several
different gravitational forces, the net force on
it is the vector sum of the individual
forces. This is called the principle of
superposition.
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Gravitational Attraction of Spherical Bodies
Gravitational force between a point mass and a
sphere the force is the same as if all the mass
of the sphere were concentrated at its center.
a consequence of 1/r2 (inverse square law)
Sphere must be radial symmetric
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Gravitational Force at the Earths Surface
The center of the Earth is one Earth radius away,
so this is the distance we use
g
...until altitude becomes comparable to the
radius of the Earth. Then the decrease in the
acceleration of gravity is much larger
The acceleration of gravity decreases slowly with
altitude...
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In the Space Shuttle
a) they are so far from Earth that Earths
gravity doesnt act any more b) gravitys force
pulling them inward is cancelled by the
centripetal force pushing them outward c) while
gravity is trying to pull them inward, they are
trying to continue on a straight-line path d)
their weight is reduced in space so the force of
gravity is much weaker

• Astronauts in the space shuttle float because

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In the Space Shuttle
a) they are so far from Earth that Earths
gravity doesnt act any more b) gravitys force
pulling them inward is cancelled by the
centripetal force pushing them outward c) while
gravity is trying to pull them inward, they are
trying to continue on a straight-line path d)
their weight is reduced in space so the force of
gravity is much weaker

• Astronauts in the space shuttle float because

Astronauts in the space shuttle float because
they are in free fall around Earth, just like a
satellite or the Moon. Again, it is gravity
that provides the centripetal force that keeps
them in circular motion.
Follow-up How weak is the value of g at an
altitude of 300 km?
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Satellite Motion FG and acp
Consider a satellite in circular motion
Gravitational Attraction
Necessary centripetal acceleration
Relationship between FG and acp will be important
for many gravitational orbit problems
not all satellite orbits are circular!
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A geosynchronous satellite is one whose orbital
period is equal to one day. If such a satellite
is orbiting above the equator, it will be in a
fixed position with respect to the ground. These
satellites are used for communications and
weather forecasting.
How high are they?
RE 6378 km ME 5.87 x 1024 kg
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Averting Disaster
a) its in Earths gravitational field b) the net
force on it is zero c) it is beyond the main pull
of Earths gravity d) its being pulled by the
Sun as well as by Earth e) none of the above

• The Moon does not crash into Earth because

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Averting Disaster
a) its in Earths gravitational field b) the net
force on it is zero c) it is beyond the main pull
of Earths gravity d) its being pulled by the
Sun as well as by Earth e) none of the above

• The Moon does not crash into Earth because

The Moon does not crash into Earth because of
its high speed. If it stopped moving, it would,
of course, fall directly into Earth. With its
high speed, the Moon would fly off into space if
it werent for gravity providing the centripetal
force.
Follow-up What happens to a satellite orbiting
Earth as it slows?
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Two Satellites
a) 1/8 b) ¼ c) ½ d) its the same e) 2

• Two satellites A and B of the same mass are
  going around Earth in concentric orbits. The
  distance of satellite B from Earths center is
  twice that of satellite A. What is the ratio of
  the centripetal force acting on B compared to
  that acting on A?

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Two Satellites
a) 1/8 b) ¼ c) ½ d) its the same e) 2

• Two satellites A and B of the same mass are
  going around Earth in concentric orbits. The
  distance of satellite B from Earths center is
  twice that of satellite A. What is the ratio of
  the centripetal force acting on B compared to
  that acting on A?

• Using the Law of Gravitation
• we find that the ratio is .

Note the 1/R2 factor
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Gravitational Potential Energy
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Energy Conservation
Total mechanical energy of an object of mass m a
distance r from the center of the Earth
This confirms what we already know as an object
approaches the Earth, it moves faster and faster.
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Escape Speed
Escape speed the initial upward speed a
projectile must have in order to escape from the
Earths gravity
from total energy
If initial velocity ve, then velocity at large
distance goes to zero. If initial velocity is
larger than ve, then there is non-zero total
energy, and the kinetic energy is non-zero when
the body has left the potential well
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Maximum height vs. Launch speed
Speed of a projectile as it leaves the Earth, for
various launch speeds
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Black holes
If an object is sufficiently massive and
sufficiently small, the escape speed will equal
or exceed the speed of light light itself will
not be able to escape the surface. This is a
black hole.
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Gravity and light
Light will be bent by any gravitational field
this can be seen when we view a distant galaxy
beyond a closer galaxy cluster. This is called
gravitational lensing, and many examples have
been found.
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Keplers Laws of Orbital Motion
Johannes Kepler made detailed studies of the
apparent motions of the planets over many years,
and was able to formulate three empirical laws
1. Planets follow elliptical orbits, with the Sun
at one focus of the ellipse.
Elliptical orbits are stable under inverse-square
force law.
You already know about circular motion...
circular motion is just a special case of
elliptical motion
Only force is central gravitational attraction -
but for elliptical orbits this has both radial
and tangential components
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Keplers Laws of Orbital Motion
2. As a planet moves in its orbit, it sweeps out
an equal amount of area in an equal amount of
time.
r
v ?t
This is equivalent to conservation of angular
momentum
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Keplers Laws of Orbital Motion
3. The period, T, of a planet increases as its
mean distance from the Sun, r, raised to the 3/2
power.
This can be shown to be a consequence of the
inverse square form of the gravitational force.
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Orbital Maneuvers
Which stable circular orbit has the higher speed?
How does one move from the larger orbit to the
smaller orbit?
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Binary systems
If neither body is infinite mass, one should
consider the center of mass of the orbital motion
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Guess My Weight
If you weigh yourself at the equator of Earth,
would you get a bigger, smaller, or similar value
than if you weigh yourself at one of the poles?
a) bigger value b) smaller value c) same
value
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Guess My Weight
If you weigh yourself at the equator of Earth,
would you get a bigger, smaller, or similar value
than if you weigh yourself at one of the poles?
a) bigger value b) smaller value c) same
value
The weight that a scale reads is the normal
force exerted by the floor (or the scale). At
the equator, you are in circular motion, so there
must be a net inward force toward Earths center.
This means that the normal force must be
slightly less than mg. So the scale would
register something less than your actual weight.
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Earth and Moon I
a) the Earth pulls harder on the Moon b) the
Moon pulls harder on the Earth c) they pull on
each other equally d) there is no force between
the Earth and the Moon e) it depends upon
where the Moon is in its orbit at that time

• Which is stronger, Earths pull on the Moon, or
  the Moons pull on Earth?

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Earth and Moon I
a) the Earth pulls harder on the Moon b) the
Moon pulls harder on the Earth c) they pull on
each other equally d) there is no force between
the Earth and the Moon e) it depends upon where
the Moon is in its orbit at that time

• Which is stronger, Earths pull on the Moon, or
  the Moons pull on Earth?

By Newtons Third Law, the forces are equal and
opposite.