Law of Universal Gravitation

1
Law of Universal Gravitation

• Holt Physics
• Pages 263 - 265

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Vertical vs horizontal circles

• Unlike object moving in horizontal circles, an
  object moving in a vertical circle is effected by
  weight.
• In vertical circles, the speed is not constant
  and neither is the tension in the string.

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Vertical vs horizontal circles

• At the top of the circular path, the speed and
  the tension in the string are at a minimum.
• At the bottom of the circle, the speed and
  tension in the string are at a maximum.

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forces acting in a vertical circle
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forces acting in a vertical circle

• At the top and bottom of the circle, FC ?FV.
• Therefore, at the top of the circle
• TS Tension in string(N)
• m mass (kg)
• g 9.8 m/s2, down
• v speed at that point (m/s)
• r radius of circle (m)

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forces acting in a vertical circle

• At the top and bottom of the circle, FC ?FV.
• Therefore, at the bottom of the circle
• TS Tension in string (N)
• m mass (kg)
• g 9.8 m/s2, down
• v speed at that point (m/s)
• r radius of circle (m)

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• The formulas you learned that apply to horizontal
  circles can also be used to apply to vertical
  circles at certain points.

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formulas
T period or time for one revolution (sec) f
frequency or revolutions per second (Hz or
sec-1)
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formulas

• T period or time for one revolution (sec)
• f frequency or revolutions per second (Hz or
  sec-1)

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formulas

• ac centripetal acceleration (m/s2)
• r radius of circle (m)
• v speed (m/s)

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formulas

• ac centripetal acceleration (m/s2)
• r radius of circle (m)
• T period or time for one revolution (sec)

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formulas

• Fc centripetal force (N)
• m mass (kg)
• ac centripetal acceleration (m/s2)

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formulas

• Fc centripetal force (N)
• m mass (kg)
• v speed (m/s)
• r radius of circle (m)

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formulas

• Fc centripetal force (N)
• m mass (kg)
• r radius of circle (m)
• T period or time for one revolution (sec)

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Sample Problem

• A stuntman swings from the end of a 4 m long rope
  along the arc of a circle. At the bottom of his
  path his speed is 9 m/s. (a) What is the
  centripetal acceleration at this point? (b) If
  his mass is 70 kg, find the tension in the rope
  at this point.

a. 20.25 m/s2
b. 2103 N
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critical velocity

• The critical velocity is the minimum velocity an
  object can have and remain in a vertical circle
  of constant radius.
• If an object does not maintain the critical
  velocity, the radius of the orbit begins to decay
  (get smaller).

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Critical velocity

• vcrit critical velocity (m/s)
• r radius (m)
• g acceleration due to gravity (m/s2)

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Sample Problem

• A carnival clown rides a motorcycle down a ramp
  and around a loop-the-loop. If the loop has a
  radius of 18 m, what is the slowest speed the
  rider can have at the top of the loop to avoid
  falling?

13.28 m/s
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Sample Problem

• A 75-kg pilot flies a plane in a loop near the
  earths surface. At the top of the loop, where
  the plane is completely upside-down for an
  instant, the pilot hangs freely in the seat and
  does not push against the seat belt. The
  airspeed indicator reads 120 m/s. What is the
  radius of the planes loop?

1469 m
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Newtons Law of Universal Gravitation

• Law of Universal Gravitation - there is a force
  of attraction between any two objects with mass.

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Newtons Law of Universal Gravitation

• F Force of attraction (N)
• G 6.67 10-11 Nm2/kg2
• m1 mass of object one (kg)
• m2 mass of object two (kg)
• d distance between the centers of the two
  objects (m)

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Accepted Values

• me mass of the Earth
• (5.98 1024 kg)
• re radius of the Earth
• (6.37 106 m)

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Satellites

• Satellites actually move in elliptical orbits but
  we will treat them as though they are circular
  orbits. The weight of the satellite is actually
  the gravitational attraction between the
  satellite and the Earth.

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Acceleration due to Gravity

• When the weight of the satellite (msg) is set
  equal to the gravitational attraction between the
  earth and the satellite (Gmsme/d2), the mass of
  the satellite will cancel out telling us that the
  acceleration due to gravity at any point can be
  calculated using the following equation.

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Acceleration due to Gravity

• G 6.67 10-11 Nm2/kg2
• m2 mass of the celestial body at the center of
  rotation (kg)
• d distance between the centers of the two
  objects (m)

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Satellite Speed

• The critical speed of a satellite can be
  calculated using

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Period of a Satellite

• The period of a satellite can be calculated using

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Weightlessness

• True weightlessness does not exist.
• In order to be truly weightless you would have to
  be infinitely far from all other objects with
  mass, since this is not possible, all objects
  have weight.

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Weightlessness

• Astronauts are said to be weightless but in
  actuality they are in freefall toward the Earth
  just as their spaceship is. Hence they are both
  accelerating at the same rate so you have the
  appearance of weightlessness.
• The misnomer (wrong name) came about due to the
  fact that if you attempted to weigh them the
  scale would register a value of zero since both
  are in free fall

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• Therefore, the astronauts would weigh zero but
  would not have a weight of zero.

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Field Concept

• A field is a region in which a suitable detector
  experiences a force.
• A suitable detector for a gravitational field is
  an object with a very small mass.