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Lecture 16
Rotational Dynamics
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Angular Momentum
p
v

F
m
 mr
t
t
t
For circular motion:

mr



  I  L


2
  rF  mr



t
t
t
t
2
 define: L  mr   mrv  angular momentum
2
Angular Momentum
Consider a particle moving in a circle
of radius r,
I = mr2
L = Iω = mr2ω = rm(rω)
= rmvt = rpt
Angular Momentum
For more general motion (not necessarily circular),
The tangential component of the
momentum, times the distance
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
Conservation of Angular Momentum
If the net external torque on a system is zero, the
angular momentum is conserved.
As the moment of inertia decreases, the
angular speed increases, so the angular
momentum does not change.
L

t
Thus,   0  L constant


Iii  I f  f or I f  f


Figure Skater
A figure skater spins with her arms
a) the same
extended. When she pulls in her arms,
b) larger because she’s rotating
she reduces her rotational inertia
faster
and spins faster so that her angular
momentum is conserved. Compared
c) smaller because her rotational
inertia is smaller
to her initial rotational kinetic energy,
her rotational kinetic energy after she
pulls in her arms must be:
Figure Skater
A figure skater spins with her arms
a) the same
extended. When she pulls in her arms,
b) larger because she’s rotating
she reduces her rotational inertia
faster
and spins faster so that her angular
momentum is conserved. Compared
c) smaller because her rotational
inertia is smaller
to her initial rotational kinetic energy,
her rotational kinetic energy after she
pulls in her arms must be:
KErot = I 2 = L  (used L = I ).
Because L is conserved, larger 
means larger KErot.
Where does the “extra” energy come from?
KErot = I 2 = L 2 (used L = I ).
Because L is conserved, larger 
means larger KErot.
Where does the “extra” energy come from?
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 Δr does the
work to raise the kinetic energy
Conservation of Angular Momentum
Angular momentum is also conserved in rotational
collisions
larger I, same total angular
momentum, smaller angular velocity
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:
τ=rF
Work is force times the distance on the arc:
W=sF
s = r Δθ
W = (r Δθ) F = rF Δθ = τ Δθ
The work-energy theorem applies as usual.
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):
Dumbbell II
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 ?
a) case (a)
b) case (b)
c) no difference
d) it depends on the rotational
inertia of the dumbbell
Dumbbell II
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.
a) case (a)
b) case (b)
c) no difference
d) it depends on the rotational
inertia of the dumbbell
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 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)
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
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
The linear momentum of components related to
the vector angular momentum of the system
Optional material
Section 11.9
Applied tangential force related to the torque vector
Optional material
Section 11.9
Applied torque over time related to change
in the vector angular momentum.
Optional material
Section 11.9
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
What is the torque (from gravity) around the supporting point?
Which direction does it point?
Without the spinning wheel: does this make sense?
With the spinning wheel: how is L changing?
Why does the wheel not fall?
Does this violate Newton’s 2nd?
Gravity
Newton’s Law of Universal Gravitation
Newton’s insight: The force accelerating
an apple downward is the same force that
keeps the Moon in its orbit.
Universal Gravitation
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.
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
Gravitational Force at the Earth’s Surface
The center of the Earth is one Earth radius away, so
this is the distance we use:
g
The acceleration of gravity
decreases slowly with altitude...
...until altitude becomes comparable to the
radius of the Earth. Then the decrease in the
acceleration of gravity is much larger:
In the Space Shuttle
a) they are so far from Earth that Earth’s gravity
doesn’t act any more
Astronauts in the b) gravity’s force pulling them inward is cancelled
by the centripetal force pushing them outward
space shuttle
c) while gravity is trying to pull them inward, they
are trying to continue on a straight-line path
float because:
d) their weight is reduced in space so the force of
gravity is much weaker
In the Space Shuttle
a) they are so far from Earth that Earth’s gravity
doesn’t act any more
Astronauts in the b) gravity’s force pulling them inward is cancelled
by the centripetal force pushing them outward
space shuttle
c) while gravity is trying to pull them inward, they
are trying to continue on a straight-line path
float because:
d) their weight is reduced in space so the force of
gravity is much weaker
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?
Satellite Motion: FG and acp
Consider a satellite in circular motion*:
Gravitational Attraction:
Necessary centripetal acceleration:
• Does not depend on mass of the satellite!
• larger radius = smaller velocity
smaller radius = larger velocity
Relationship between FG and acp will be important
for many gravitational orbit problems
*
not all satellite orbits are circular!
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
Averting Disaster
a) it’s in Earth’s gravitational field
b) the net force on it is zero
The Moon does not
c) it is beyond the main pull of Earth’s gravity
crash into Earth
d) it’s being pulled by the Sun as well as by
Earth
because:
e) none of the above
Averting Disaster
a) it’s in Earth’s gravitational field
b) the net force on it is zero
The Moon does not
c) it is beyond the main pull of Earth’s gravity
crash into Earth
d) it’s being pulled by the Sun as well as by
Earth
because:
e) none of the above
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 weren’t for gravity providing the
centripetal force.
Follow-up: What happens to a satellite orbiting Earth as it slows?
Two Satellites
Two satellites A and B of the same mass
are going around Earth in concentric
orbits. The distance of satellite B from
Earth’s center is twice that of satellite A.
What is the ratio of the centripetal force
acting on B compared to that acting on A?
a) 1/8
b) ¼
c) ½
d) it’s the same
e) 2
Two Satellites
Two satellites A and B of the same mass
are going around Earth in concentric
orbits. The distance of satellite B from
Earth’s 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 .
a) 1/8
b) ¼
c) ½
d) it’s the same
e) 2
Note the
1/R2 factor
Gravitational Potential Energy
Gravitational potential energy of an object of mass
m a distance r from the Earth’s center:
(U =0 at r -> infinity)
Very close to the Earth’s
surface, the gravitational
potential increases linearly
with altitude:
Gravitational potential energy, just like all
other forms of energy, is a scalar. It
therefore has no components; just a sign.
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.
Escape Speed
Escape speed: the initial upward speed a
projectile must have in order to escape from
the Earth’s 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
Maximum height vs. Launch speed
Speed of a projectile as it leaves the Earth,
for various launch speeds
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.
The light itself has mass (in the
mass/energy relationship of Einstein), or
spacetime itself is curved
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.
Kepler’s 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
Kepler’s 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 Δt
This is equivalent to conservation of angular momentum
Kepler’s 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.
Orbital Maneuvers
Which stable circular orbit has
the higher speed?
How does one move from the
larger orbit to the smaller orbit?
Binary systems
If neither body is “infinite” mass, one should consider
the center of mass of the orbital motion
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
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 Earth’s center. This
means that the normal force must be slightly less than mg. So the
scale would register something less than your actual weight.
Earth and Moon I
a) the Earth pulls harder on the Moon
Which is stronger,
b) the Moon pulls harder on the Earth
Earth’s pull on the
c) they pull on each other equally
Moon, or the
d) there is no force between the Earth and
the Moon
Moon’s pull on
Earth?
e) it depends upon where the Moon is in its
orbit at that time
Earth and Moon I
a) the Earth pulls harder on the Moon
Which is stronger,
b) the Moon pulls harder on the Earth
Earth’s pull on the
c) they pull on each other equally
Moon, or the
d) there is no force between the Earth and
the Moon
Moon’s pull on
Earth?
e) it depends upon where the Moon is in its
orbit at that time
By Newton’s Third Law, the forces
are equal and opposite.