Physics 201: Lecture 21

1
Physics 201 Lecture 21

• Universal Law of Gravitation
• Newtons apple and moon
• Weight
• Escape velocity
• Orbits of satellites

2
Newtons Law of Gravitation

• Every particle in the universe attracts every
  other particle with a force along the line
  joining them. The force is directly proportional
  to the product of their masses and inversely
  proportional to the square of the distance
  between them.

Note particle! If an object is extended you
must treat the vector sum of all the forces due
to all the particles making up the object. This
is done automatically by considering the object
as if all its mass were concentrated at the
center of mass (or the center of gravity!?) If
you are considering a system of extended objects,
you must still consider the center of mass.
3
Newtons Apple and Moon

• Newton guessed that the acceleration of the
  moon in its orbit had the same cause as the
  acceleration of an apple on the surface.

The acceleration of the Moon is
4
Universal Law of Gravitation

• This is an example of an inverse square law
• The magnitude of the force varies as the inverse
  square of the separation of the particles
• The law can also be expressed in vector form

• F12 is the force exerted by particle 1 on
  particle 2
• The negative sign in the vector form of the
  equation indicates that particle 2 is attracted
  toward particle 1
• F21 is the force exerted by particle 2 on
  particle 1

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Central force
Direction along the line joining the center --
attractive
Torque?? We know that there is no torque on an
object by the
gravitational force about the center of mass.
No torque means conservation of angular momentum.
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Potential Energy
Work done to bring mass m from initial to final
position.
Zero point is arbitrary. Choose zero at infinity.
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Gravity...

• Near the Earths surface
• R12 RE
• Wont change much if we stay near the Earth's
  surface.
• since RE gtgt h, RE h RE.

RE
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Gravity...
Leaky Cup

• Near the Earths surface...

???
g

• So Fg mg ma
• a g

All objects accelerate with acceleration g,
regardless of their mass!
Where
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Variation of g with Height
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Question

• Suppose you are standing on a bathroom scale in
  y\our dorm room and it says that your weight is
  W. What will the same scale say your weight is
  on the surface of the mysterious Planet X ?
• You are told that RX 20 REarth and MX 300
  MEarth.
• (a) 0.75 W (b) 1.5 W (c)
  2.25 W

E
X
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Solution

• The gravitational force on a person of mass m by
  another object (for instance a planet) having
  mass M is given by

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Potential Energy Near the Surface

• Consider an object of mass m near the surface of
  a planet e.g. Earth.

2
1
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Escaping Gravity

• Kinetic energy of the object must be greater than
  its gravitational potential energy
• This defines the minimum velocity to escape
• KEPE constant
• Consider case when speed is just sufficient to
  escape to infinity with vanishing final velocity
• At infinity, KEPE0, therefore, on Earth,

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Question

• 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 acceleration of B to that of A?

1/4 1/2 0.707 1.0
Since the only force is the gravitational force,
it must scale as the inverse square of their
distances from the center of the Earth.
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Measuring G

• G was first measured by Henry Cavendish in 1798
• The apparatus shown here allowed the attractive
  force between two spheres to cause the rod to
  rotate
• The mirror amplifies the motion
• It was repeated for various masses

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Lecture 21 Preflight 1

• Superman stands on the top of Mount Everest and
  throws a baseball horizontally with a speed such
  that the baseball goes into a circular orbit
  around the Earth. When the baseball is in orbit,
  its acceleration
• depends on how fast the baseball is thrown.
• is zero because the ball does not fall to the
  ground.
• is slightly less than 9.8 m/s2
• is equal to 9.8 m/s2

Acceleration g is slightly smaller at the top of
the Mount Everest, due to its height.
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Lecture 21 Preflight 2

• The Moon remains in orbit around the Earth rather
  than falling into the Earth because
• It is outside the gravitational influence of the
  Earth.
• It is in balance with the gravitational forces
  from the Sun and the other planets.
• The net force on the Moon is zero
• All of these
• None of these

Because, the Moons centripetal force is equal to
the force due to the Earths gravitational pull.
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Lecture 21 Preflight 3

• A planet has two moons in circular orbits around
  it. Moon 1 has mass M and is at radius r. Moon 2
  has mass 2M and is at a radius 2r. The magnitude
  of the gravitational force exerted by the moon 2
  on the planet is
• Four times as large as for moon 1.
• Twice as large as for moon 1.
• Equal to that for moon 1.
• Half as large as for moon 1.
• One-fourth as large as for moon 1.

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Lecture 21 Preflight 4

• A new planet has been found to orbit the Sun. the
  new planet has an orbital period that is
• Greater than a year.
• Equal to an year.
• Less than a year.
• Need more information.

If a new planet is found it has to have an orbit
that is larger than the known planets. Therefore,
its period will be longer.
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Lecture 21 Preflight 5

• Suppose you are standing in the space station in
  orbit around the Earth at an altitude of 350 km
  above the surface and you let go your helmet.
• It drops to the floor of the station very slowly.
  
• It remains near your hand because there is no
  gravity in orbit.
• It moves to the outside surface because of
  centrifugal force.
• None of the reasons above.

There is still gravity acting on the helmet. It
continues to be balanced by its centripetal force
while in orbit.
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Lecture 21 Preflight 6

• A 1000 kg object is sent to the Moon. At almost
  the same time another object of 3000 kg is also
  sent. They both have the least possible energy to
  make the trip. Compared to the energy needed by
  the lighter object, the heavier object needs
• Four times the energy.
• Three times the energy.
• The same energy.
• Twelve times the energy.
• Half the energy.