T' K' Ng, HKUST

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HK IPhO Training class (mechanics)
T. K. Ng, HKUST
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Lecture III
(1)Reference frame problem Coriolis
force (2)Circular motion and angular
momentum (3)Planetary motion (Keplers Laws)
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Recall Newtons First LawAn object remains in
its state of rest or uniform motion (moving with
uniform velocity) in a straight line if there is
no net force acting on it.
(1)Reference frame problem, coriolis force.
F 0
The first part of the first law is probably easy
to accept. However, the second part is not so
trivial.
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An astronaut at an orbit around the earth
(although the astronaut is accelerating, he/she
does not feel force!).
It is trickier. It is related to the problem of
so-called inertial reference frame
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Reference frame problem Newtons first law
cannot be simultaneously correct for all
observers (with different motions).
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I have a question.
Question Suppose you are sitting in a rotating
Merry-go-around. Do you think that Newtons
law will be correct? For example, is the net
force acting on a body zero when it is not moving?
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Problem Motion is defined only relative to a
coordinate system
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You are on the platform. You can see the
passengers like this
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Now you are on the train. You will see the
passengers
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Like this.
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Two observers moving with acceleration with
respect to each other and looking at an object at
rest with the first observers.
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Two observers moving with acceleration with
respect to each other and looking at an object at
rest with the first observers.
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To the first observer, the force acting on the
object is zero according to Newtons Law since it
is at rest.
a
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To the second observer, the object is
accelerating, so there must be force acting on
the object according to Newtons Law!
a
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To the second observer, the object is
accelerating, so there must be force acting on
the object according to Newtons Law!
a
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Who is correct!?
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The reference frames where Newtons first law
holds are called inertial frames. Roughly
speaking, inertial frames are those which are not
accelerating (or decelerating) themselves.
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Transformation between 2 reference frames
Let
Coriolis force
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In the example of astronaut going around the
earth, people on the earth see that the astronaut
is subjected to gravitational force and is
accelerating towards the center of earth.
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In the example of astronaut going around the
earth, people on the earth see that the astronaut
is subjected to gravitational force and is
accelerating towards the center of earth.
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In the example of astronaut going around the
earth, people on the earth see that the astronaut
is subjected to gravitational force and is
accelerating towards the center of earth.
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?
?
?
However, the astronaut himself/herself is
accelerating and is not in an inertial frame.
Therefore, he/she does not find Newtons first
law to be correct!
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II. Circular motion Angular Momentum
circular motion (car turning in circle, planets
around sun with constant speed)
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Newtons Law ? a radial attractive force between
two body is needed to sustain circular motion of
one body around another.
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Why?
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Why?
Direction of acceleration in circular motion
pointing towards center of circle
now
? radial attractive force needed from F ma !
later
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Mathematics of Circular motion (constant speed)
x(t)rcos(?t), y(t) rsin(?t) show
that vx(t) -vosin(?t), vy(t) vocos(?t) ax(t)
-aocos(?t), ay(t) -aosin(?t). What are vo,
ao?
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Mathematics of Circular motion (constant speed)
Show that to sustain circular motion, there must
exist a force Fmr?2 pointing towards center of
circle according to Newtons Law.
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An example of circular motion
Earth moving around sun under gravitational force
Exercise Derive the relation between velocity
and r in this case.
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Some other examples
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(1) Objects on an Merry-go-around tend to fly
outside unless bound by a radial force (or
motion stopped by friction).
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(2) Bicycles in circular motion dont fall even
if it is not standing straight
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There are two forces acting on the person
reaction force ( along the bicycle )
gravitational force (acting down)
radial force that supports circular motion of
bicycle.


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Angular Momentum
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Cross-Product
Or
Right Hand Rule
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Cross-Product
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y
For Circular motion
x
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y
For Circular motion
x
Notice, angular momentum is a constant vector
for circular motion
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Angular Momentum
This is an example of conservation of angular
momentum.
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Angular Momentum
In general, angular momentum is nonzero when an
object is not moving along a straight line , i.e.
it is a measure of rotation!
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Angular Momentum
How about the following trajectory?
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Energy in circular motion
In general Total EnergyP.E. K.E. For
gravitational force,
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Energy in circular motion
Notice that both P.E. and K.E. are constants of
motion for circular motion (r remains unchanged)
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Actually Circular motion is just one possible
realization of motion of objects under central
force
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Actually Circular motion is just one possible
realization of motion of objects under central
force
In general, objects can move in many different
ways under central force,
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We shall consider one particular example
planetary motion in the following.
In general, objects can move in many different
ways under central force,
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II. Planetary Motion.(Motion of (point) object
under gravitational force)
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a. Conservation of angular momentum for objects
moving under central force.
(polar coordinate)
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Mathematics of angular momentum conservation
(conservation of angular momentum)
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In particular, angular momentum of planets moving
around sun are conserved
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b. Plausible (planetary) orbits with
(polar coordinate)
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Notice both closed (ellipse) and open
(hyperbola) orbits exist!
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(circle)
(ellipse)
(parabola)
(hyperbola)
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Keplers Law for closed orbits

• Law of Orbits Planets move in elliptical orbits
  with the sun at one focus
• Law of Areas A line joining any planet to the
  sun sweeps out equal areas in equal intervals of
  time
• Law of Periods The square of the period of
  revolution of any planet is proportional to the
  cube of the semimajor axis a of the orbit.

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Keplers Law for closed orbits

• Law of Orbits consequence of gravitational force
  (shall not prove)
• Law of Areas consequence of conservation of
  angular momentum
• Law of Periods consequence of (1) (2)
  conservation of energy.

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Keplers 2nd Law

• Consider the area ?A swap out by the orbit at a
  small time interval.

? r
r
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Keplers 2nd Law

• L is conserved ? ?A ? ? t! (A line joining any
  planet to the sun sweeps out equal areas in equal
  intervals of time)

? r
r
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Keplers 3rd Law

• Consider the rate of change of area ?A/? t at the
  two extreme points A, B

c
A
B
a
(Semi-major axis)
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Keplers 3rd Law

• Consider the rate of change of area ?A/? t at the
  two extreme points A, B

c
A
B
a
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Keplers 3rd Law

• Consider the rate of change of area ?A/? t at the
  two extreme points A, B

c
A
B
a
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Keplers 3rd Law

• Next we consider the total energies at points A
  and B

Conservation of energy
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Keplers 3rd Law

• Eq.(1)Eq.(2)Eq.(3) ?

Substitute in Eq.(2)?
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Keplers 3rd Law

• The rate of change of area ?A/? t at the extreme
  points B is

c
A
B
a
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Keplers 3rd Law

• Using property of ellipse a2b2c2

b
c
A
B
a
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Keplers 3rd Law

• The area of an ellipse is A?ab. Therefore, the
  period of the orbit is

Keplers Third Law
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Application of Keplers Law conservation
Laws Notice the importance of knowing the
geometrical properties of ellipse
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