PHYS 1443-001, Summer I 2005

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PHYS 1443 Section 001Lecture 7
Monday June 13, 2005 Dr. Andrew Brandt

• Friction
• Resistive Forces
• Newtons Law of Gravitation
• Keplers Laws

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Announcements

• Test 1
• Test 2 Thursday 6/16
• Homework
• HW3 on ch4 due Monday 6/13 at 2pm
• HW4 on ch5 due Tuesday 6/14 at 8pm
• HW5 on ch 6 due Wednesday 6/15 at 6pm

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Motion in Resistive Forces
A medium can exert resistive forces on an object
moving through it due to viscosity or other types
of frictional properties of the medium.
Some examples?
Air resistance, viscous force of liquid, etc
These forces are exerted on moving objects in the
opposite direction of the motion.
These forces are proportional to such factors as
speed. They almost always increase with
increasing speed.

• Two different cases of proportionality
• Forces linearly proportional to speed Slowly
  moving or very small objects
• Forces proportional to square of speed Large
  objects w/ reasonable speed

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Resistive Force Proportional to Speed
Since the resistive force is proportional to
speed, we can write Rbv.
Lets consider a ball of mass m falling through a
liquid.
ma
m
The above equation also tells us that as time
goes on the speed will increase and the
acceleration will decrease and eventually go to
0.
What does this mean?
An object moving in a viscous medium will obtain
speed to a certain speed (terminal speed) and
then maintain the same speed without any more
acceleration.
What is the terminal speed in above case?
The time needed to reach 63.2 of the terminal
speed is defined as the time constant, tm/b.
How do the speed and acceleration depend on time?
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Newtons Law of Universal Gravitation
People have been very curious about the stars in
the sky, and have made observations over
centuries. But the data people collected were
not adequately explained until Newton discovered
the law of gravitation.
Every particle in the Universe attracts every
other particle with a force that is directly
proportional to the product of their masses and
inversely proportional to the square of the
distance between them.
With G
How would you write this principle mathematically?
G is the universal gravitational constant, and
its value is
Unit?
This constant is not given by the theory but must
be measured by experiment.
This form of force is known as an inverse-square
law, because the magnitude of the force is
inversely proportional to the square of the
distance between the objects.
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More on Law of Universal Gravitation
Consider two particles exerting gravitational
forces on each other.
Two objects exert gravitational force on each
other following Newtons 3rd law.
What do you think the negative sign means?
It means that the force exerted on the particle 2
by particle 1 is attractive force, pulling mass
2 towards mass 1.
Gravitational force is a field force Field
forces act on objects without physical contact
between the objects and independent of the medium
between them.
How would you describe the gravitational force
on the surface of the Earth?
The gravitational force exerted by a finite size,
spherically symmetric mass distribution on a
particle outside the distribution is the same as
if the entire mass of the distribution was
concentrated at its center.
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Free Fall Acceleration Gravitational Force
Weight of an object with mass m is mg. Using the
force exerted on a particle of mass m located on
the surface of the Earth, one can get
What would the gravitational acceleration be if
the object is at an altitude h above the surface
of the Earth?
What does this tell us about the gravitational
acceleration?

• The gravitational acceleration is independent of
  the mass of the object
• The gravitational acceleration decreases as the
  altitude increases
• If the distance from the surface of the Earth
  gets infinitely large, the weight of the object
  approaches 0.

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Example for Gravitation
Using the fact that g9.80m/s2 at the Earths
surface, find the average density of the Earth.
Since the gravitational acceleration is
Solving for ME
Therefore the density of the Earth is
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Example for Gravitational Force
The international space station is designed to
operate at an altitude of 350km. When completed,
it will have a weight (measured on the surface of
the Earth) of 4.22x106N. What is its weight when
in its orbit?
The total weight of the station on the surface of
the Earth is
Since the orbit is at 350km above the surface of
the Earth, the gravitational force at that height
is
Therefore the weight in orbit is
Note this is only 10 less. Why?
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The Law of Gravity and the Motion of Planets

• Newton assumed that the law of gravitation
  applies the same whether it is acting on the Moon
  or on an apple on the surface of the Earth. The
  interacting bodies are assumed to be point like
  particles.

Newton predicted that the ratio of the Moons
acceleration aM to the apples acceleration g
would be
Therefore the centripetal acceleration of the
Moon, aM, is
Note this acceleration is not the acceleration of
an object on the surface of the Moon, but rather
the acceleration of the moon towards the Earth
Newton also calculated the Moons orbital
acceleration aM from the knowledge of its
distance from the Earth and its orbital period,
T27.32 days2.36x106s
This means that the Moons distance is about 60
times that of the Earths radius, its
acceleration is reduced by the square of the
ratio. This proves that the inverse square law
is valid.
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Keplers Laws Ellipse
Ellipses have two different axis, major (long)
and minor (short) axis, and two focal points, F1
F2 a is the length of a semi-major axis b is
the length of a semi-minor axis
Kepler lived in Germany and discovered the laws
governing planets movement some 70 years before
Newton, by analyzing data.

1. All planets move in elliptical orbits with the
   Sun at one focal point.
2. The radius vector drawn from the Sun to a planet
   sweeps out equal area in equal time intervals.
   (Angular momentum conservation)
3. The square of the orbital period of any planet is
   proportional to the cube of the semi-major axis
   of the elliptical orbit.

Keplers laws can be derived from Newtons laws.
We will show that Keplers third law is the
direct consequence of the inverse square nature
of the law of gravitation.
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Keplers Second Law and Angular Momentum
Conservation
Warning This slide contains many concepts we
have not covered yet
Consider a planet of mass Mp moving around the
Sun in an elliptical orbit.
Since the gravitational force acting on the
planet is always directed radially, it is a
central force
Therefore the torque acting on the planet from
this force is always 0.
Since torque is the time rate of change of
angular momentum L, the angular momentum is
constant.
Because the gravitational force exerted on a
planet by the Sun results in no torque, the
angular momentum L of the planet is constant.
Since the area swept by the motion of the planet
is
This is Kepers second law which states that the
radius vector from the Sun to a planet sweeps out
equal areas in equal time intervals.
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Keplers Third Law
It is important to show that Kepers Third Law
can be predicted from the inverse square law for
circular orbits.
Since the gravitational force exerted by the Sun
is radially directed toward the Sun to keep the
planet in a (roughly) circular orbit, we can
apply Newtons second law
Since the orbital speed, v, of the planet with
period T is
The above can be written
Solving for T one can obtain
with
This is Keplers third law. Its also valid for
ellipse with ra, the length of the semi-major
axis. The constant Ks is independent of the mass
of the planet.
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Example of Keplers Third Law
Calculate the mass of the Sun using the fact that
the period of the Earths orbit around the Sun is
3.16x107s, and its distance from the Sun is
1.496x1011m.
Using Keplers third law.
The mass of the Sun, Ms, is