PHYS 1443501, Spring 2004

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PHYS 1443 Section 501Lecture 12
Wednesday, Mar 3, 2004 Dr. Andrew Brandt

• Newtons Law of Gravitation and Keplers Laws
• Work Done by Constant Force
• Work Done by Varying Force
• Kinetic Energy

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Announcements

• HW5 on Ch. 6 is due Mon. March 8 at midnight
• Test Solutions available
• Interim grades posted on website
• If you dont like your grade there is still
  time to change it it starts with the HWs, spend
  more time on them and everything else will follow

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Newtons Law of Universal 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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Free Fall Acceleration Gravitational Force
Weight of an object with mass m is mg. Using the
force exerting on a particle of mass m 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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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

• All planets move in elliptical orbits with the
  Sun at one focal point.
• The radius vector drawn from the Sun to a planet
  sweeps out equal area in equal time intervals.
  (Angular momentum conservation)
• 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.
Keplers third law is the direct consequence of
the inverse square nature of the law of
gravitation.
quizexample
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Work Done by a Constant Force
Work in physics is done only when a sum of
forces exerted on an object causes motion of the
object.
Free Body Diagram
M
M
Which force did the work?
How much work did it do?
Unit?
Physical work is done only by the component of
the force along the movement of the object.
What does this mean?
Work is energy transfer!!
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Example of Work w/ Constant Force
A man cleaning a floor pulls a vacuum cleaner
with a force of magnitude F50.0N at an angle of
30.0o with East. Calculate the work done by the
force on the vacuum cleaner as it is displaced by
3.00m to East.
M
M
Does work depend on mass of the object being
worked on?
Yes
Why dont I see the mass term in the work at all
then?
It is reflected in the force. If the object has
smaller mass, its would take less force to move
it the same distance as the heavier object. So
it would take less work.
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Scalar (Dot) Product of Two Vectors

• Product of magnitude of the two vectors and the
  cosine of the angle between them

• Operation is commutative

• Operation follows distribution law of
  multiplication

• Scalar products of Unit Vectors

• How does scalar product look in terms of
  components?

Is this a vector or a scalar?
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Example of Work by Scalar Product
A particle moving in the xy plane undergoes a
displacement d(2.0i3.0j)m as a constant force
F(5.0i2.0j) N acts on the particle.
a) Calculate the magnitude of the displacement
and that of the force.
b) Calculate the work done by the force F.
Can you do this using the magnitudes and the
angle between d and F?
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Work Done by a Varying Force

• If the force depends on the position of a
  particle during the motion
• one must consider the work increment in a small
  segment of the position where the force can be
  considered constant

• Then add them all up throughout the entire motion
  (xi? xf)

In the limit where Dx?0

• If more than one force is acting, the net work is
  done by the net force

An example of a force that depends on position is
the spring force
The work done by the spring force is
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Kinetic Energy and Work-Kinetic Energy Theorem

• Some problems are hard to solve using Newtons
  second law
• If forces exerted on the object during the motion
  are complicated
• Relate the work done on the object by the net
  force to the change of the speed of the object

Suppose a net force SF was exerted on an object
over a displacement d to increase its speed from
vi to vf.
M
M
The work on the object by the net force SF is
Acceleration
Kinetic Energy
Work
The work done by the net force causes a change of
objects kinetic energy.
Work