The Law of Universal Gravitation

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The Law of Universal Gravitation
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What do we know about gravity?

• Gravity is a force which exists between the Earth
  and the objects on or near it.
• The force of gravity causes a ball tossed upward
  to slow down on the way up and speed up on the
  way down.
• This change of velocity is the acceleration due
  to gravity (g).
• On or near Earth's surface, g 9.8 m/s2.
• It is the same acceleration value for all
  objects, regardless of their mass (and assuming
  that the only significant force is gravity).

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Gravity and the Motion of Planets

• In the early 1600s, German mathematician and
  astronomer Johannes Kepler mathematically
  analyzed known astronomical data from his mentor.
• Through inductive reasoning, Kepler developed
  three laws to describe the motion of planets
  about the sun.

We will study Keplers Laws of Planetary Motion
later
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What makes the planets orbit the sun (and the
moon orbit the earth?)

• Kepler couldnt explain why the planets orbited
  the sun. He said they were somehow magnetically
  driven by the sun.
• Enter Isaac Newton

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Circular Motion Requires a Centripetal Force

• Newton knew that, for the motion of the moon in a
  circular path around the Earth, there must be a
  centripetal force.
• Circular motion is clearly a departure from the
  inertial paths (straight-line) of objects.
• An unbalanced force (NET FORCE) must exist.
• Since the moon moves around the Earth in an
  approximately circular path, the NET FORCE must
  be directed INWARD, TOWARD THE EARTH.

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Newton makes a mental leap

• Whether it is a myth or a reality, the fact is
  certain that it was Newton's ability to relate
  the cause for heavenly motion (the orbit of the
  moon about the Earth) to the cause for Earthly
  motion (the fall of an apple to the Earth) which
  led him to his notion of universal gravitation.
• But how did he make such a bold connection?

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Newtons Thought Experiment

• Newton knew that the Earths curvature was such
  that for 8000 meters horizontally, the ground
  dropped 5 m vertically.

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Newtons Thought Experiment

• In the absence of gravity a projectile would move
  in a straight line path tangent to the Earth. In
  the absence of any unbalanced force, an object in
  motion will continue in motion with the same
  speed and in the same direction (the law of
  inertia).

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Newtons Thought Experiment

• Suppose that a very powerful cannon was mounted
  on top of a very tall mountain. Ignore air
  resistance.
• At launch speeds less than 8000 m/s, the
  cannonball falls to Earth.

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Newtons Thought Experiment

• At a launch speed equal to 8000 m/s, the
  cannonball follows a circular path.
• At launch speeds greater than 8000 m/s, the
  cannonball follows an elliptical path.

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Next

• Now Newton needed to come up with evidence that
  the force that acts on objects at the Earths
  surface is the same that acts on heavenly
  bodies.
• The key to this was showing how the effect of
  gravity is diluted by distance.

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• Newton knew that earthbound objects (such as
  falling apples) accelerate towards the earth at a
  rate of 9.8 m/s2.
• It was also known that the moon accelerated
  towards the earth at a rate of 0.00272 m/s2.
• If the same force causes both of these motions,
  why then is the acceleration of the moon is so
  much smaller than the acceleration of the apple?
• What is it about the force of gravity which
  causes the more distant moon to accelerate at a
  rate of acceleration which is approximately
  1/3600th the acceleration of the apple?

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An object on the Earths surface experiences a
gravitational acceleration 3600 times greater
than the moons centripetal acceleration in its
orbit of Earth. Why the big difference?
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The moon in its orbit about the earth is
approximately 60 times further from the earth's
center than the apple is.
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The Inverse Square Relationship

• The force of gravity between the earth and any
  object is inversely proportional to the square of
  the distance which separates that object from the
  earth's center.
• The moon, being 60 times further away than the
  apple, experiences a force of gravity which is
  1/(60)2 times that of the apple. The force of
  gravity follows an inverse square law.

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Its an inverse square!

• The relationship between the force of gravity
  (Fgrav) between the earth and any other object
  and the distance which separates their centers
  (d) can be expressed by the following
  relationship

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Check Your Understanding

• 1. Suppose that two objects attract each other
  with a force of 16 units. If the distance between
  the two objects is doubled, what is the new force
  of attraction between the two objects?
• 2. Suppose that two objects attract each other
  with a force of 16 units. If the distance between
  the two objects is tripled, then what is the new
  force of attraction between the two objects?

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Check Your Understanding

• 3. Suppose that two objects attract each other
  with a force of 16 units. If the distance between
  the two objects is reduced in half, then what is
  the new force of attraction between the two
  objects?
• 4. Suppose that two objects attract each other
  with a force of 16 units. If the distance between
  the two objects is reduced by a factor of 5, then
  what is the new force of attraction between the
  two objects?

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Check Your Understanding

• 5. If you wanted to make a profit in buying gold
  by weight at one altitude and selling it at
  another altitude for the same price per weight,
  should you buy or sell at the higher altitude
  location? What kind of scale must you use for
  this work?

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The Answers

• 1. 4 units
• 2. 1.78 units
• 3. 64 units
• 4. 400 units
• 5. Buy at high altitude, where the gold weighs
  less, and sell at low altitude, where the gold
  weighs more.

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To understand the inverse square relationship

• Consider the butter gun used by a busy restaurant
  to butter toast.
• A piece of toast placed 1 ft. away is covered
  with a layer of butter 1mm thick.
• If the toast is held twice as far away, how thick
  is the butter?
• If the toast is held three feet away, how thick
  is the butter?

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The Butter Gun
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What else affects gravitational force and
acceleration?

• The force which caused the apple's acceleration
  (gravity) must be dependent upon the mass of the
  apple (2nd Law).
• Since the force acting to cause the apple's
  downward acceleration also causes the earth's
  upward acceleration (3rd Law), that force must
  also depend upon the mass of the earth.

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In Summary

• The force of gravity acting between the earth and
  any other object is
• directly proportional to the mass of the earth
• directly proportional to the mass of the object
• inversely proportional to the square of the
  distance which separates the centers of the earth
  and the object

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The Law of Universal Gravitation
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What it means
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The Law of Universal Gravitation

• Nearly a century later, Lord Henry Cavendish
  determined the value of the universal gravitation
  constant using a torsion balance.
• Thus, the law could be expressed as an equation

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The Cavendish Experiment
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Gee, thats small!

• The value of G is an extremely small numerical
  value.
• Its smallness accounts for the fact that the
  force of gravitational attraction is only
  appreciable for objects with large mass.
• Knowing the value of G allows us to calculate the
  force of gravitational attraction between any two
  objects of known mass and known separation
  distance.

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Calculating Fgrav

• Determine the force of gravitational attraction
  between the earth (m 5.98 x 1024 kg) and a 70
  kg physics student if the student is standing at
  sea level, a distance of 6.37 x 106 m from
  earth's center.

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The solution is as follows
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Calculating Fgrav

• Determine the force of gravitational attraction
  between the earth (m 5.98 x 1024 kg) and a 70
  kg physics student if the student is in an
  airplane at 40000 feet above earth's surface.
  This would place the student a distance of 6.38 x
  106 m from earth's center.

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The solution is as follows
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Its Universal!

• Newton's place in the Gravity Hall of Fame is not
  due to his discovery of gravity, but rather due
  to his discovery that gravitation is universal.
• Newton's Law of Universal Gravitation extends
  gravity beyond Earth. It is about the
  universality of gravity.
• ALL objects attract ALL OTHER OBJECTS with a
  force of gravitational attraction.

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Its Universal!

• So as you sit in your seat in the physics
  classroom, you are gravitationally attracted to
  your lab partner, to the desk you are working at,
  and even to your pen or pencil.
• ALL objects attract in proportion to the product
  of their masses.
• Most gravitational forces are too minimal to be
  noticed.
• Gravitational forces only are recognizable as the
  masses of objects become large.

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Check Your Understanding

• Suppose that you have a mass of 70 kg (equivalent
  to a 154-pound person). How much mass would
  another object have to have in order for your
  body and the object to attract each other with a
  force of 1 Newton when separated by 10 meters?
• Answer m2 2.14 x 1010 kg
• Thats 23 million tons!

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The value of g

• We have often calculated the force of gravity
  (Fgrav) with which an object of mass m was
  attracted to the earth using Fgrav mg
• Now a second equation has been introduced for
  calculating the force of gravity with which an
  object is attracted to the earth.

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g varies with distance

• To understand why the value of g is so location
  dependent, we will use the two equations above to
  derive an equation for the value of g.
• First, both expressions for the force of gravity
  are set equal to each other.

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Divide both sides by m

• The acceleration due to gravity is dependent upon
  the mass of the Earth (approx. 5.98x1024 kg) and
  the distance (d) that an object is from the
  center of the Earth.
• If the value 6.38x106 m (a typical Earth radius
  value) is used for the distance from Earth's
  center, then g will be calculated to be 9.8 m/s2.

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So g depends on distance

• If an object were moved to a location which is
  two earth-radii from the center of the earth -
  that is, two times 6.38x106 m - then a
  significantly different value of g will be found.

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Golly g!

• The value of g varies inversely with the square
  of the distance from the center of the Earth.
• Just like Fgrav, g follows an inverse square law.
• g 1 / d2

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Its Universal!

• The same equation used to determine the value of
  g on Earth can also be used to determine the
  acceleration of gravity on the surface of other
  planets.
• The value of g on any other planet can be
  calculated from the mass of the planet and the
  radius of the planet.

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