CIRCULAR MOTION AND GRAVITATION

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CHAPTER 6 CIRCULAR MOTION AND GRAVITATION
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Goals for Chapter 6

• To understand the dynamics of circular motion.
• To study the unique application of circular
  motion as it applies to Newtons Law of
  Gravitation.
• To study the motion of objects in orbit as a
  special application of Newtons Law of
  Gravitation.

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Uniform circular motion is due to a centripetal
acceleration This aceleration is always pointing
to the center This aceleration is due to a net
force
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• Period the time for one revolution
• Circular motion in horizontal plane
• - flat curve
• - banked curve
• - rotating object
• 2) Circular motion in vertical plane

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Rounding a flat curve

• The centripetal force coming only from tire
  friction.

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Rounding a banked curve

• The centripetal force comes from friction and a
  component of force from the cars mass

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Dynamics of a Ferris Wheel
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The "Giant Swing" at a county fair consists of a
vertical central shaft with a number of
horizontal arms attached at its upper end. Each
arm supports a seat suspended from a 5 m long
cable, the upper end of which is fastened to the
arm at a point 3m from the central shaft. Find
the time of one revolution of the swing if the
cable supporting the seat makes an angle of 300
with the vertical.
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GRAVITATION
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Spherically symmetric objects interact
gravitationally as though all the mass of each
were concentrated at its center
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Cavendish Balance

• The slight attraction of the masses causes a
  nearly imperceptible rotation of the string
  supporting the masses connected to the mirror.
• Use of the laser allows a point many meters away
  to move through measurable distances as the angle
  allows the initial and final positions to
  diverge.

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Newtons Law of Gravitation

• Always attractive.
• Directly proportional to the masses involved.
• Inversely proportional to the square of the
  separation between the masses.
• Masses must be large to bring Fg to a size even
  close to humanly perceptible forces.

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A diagram of gravitational force
G 6.674x10-11 N.m2/kg2
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Each mass is 2 kgFind the magnitude of the net
gravitational force on each mass and its
direction
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Each mass in the figure below is 3 kg. Find the
force (magnitude and direction) on each mass in
the figure .
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WEIGHT
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Gravitational force falls off quickly

• If either m1 or m2 are small, the force
  decreases quickly enough for humans to notice.

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• In January 2005 the Huygens probe landed on
  Saturn's moon Titan, the only satellite in the
  solar system having a thick atmosphere. Titan's
  diameter is 5150 km, and its mass is 1.351023
  kg, The probe weighed 3120 N on earth. What did
  it weigh on the surface of Titan?

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Satellite Motion
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What happens when velocity rises?

• When v is large enough, you achieve escape
  velocity.

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The principle governing the motion of the
satellite is Newtons second law the force is F,
and the acceleration is v2/r, so the equation
Fnet ma becomes GmmE/r 2 mv 2/r
v GmE/r T 2pr/v
(2pr3/2)/ Larger orbits correspond to slower
speeds and longer periods.


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A 320 kg satellite experiences a gravitational
force of 800 N. What is the radius of the of the
satellites orbit? What is its altitude? F
GmEmS/r 2 r 2 GmEmS/ F r 2 (6.67 x 10 -11
N.m2/kg2) (5.98 x 10 24 kg) (320 kg ) / 800 N r
2 1.595 x 1014 m2 r 1.26 x 107 m Altitude
1.26 x 107 m radius of the Earth Altitude
1.26 x 107 m 0.637 x 107 0.623 x 107 m
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We want to place a satellite into circular orbit
300km above the earth surface. What speed, period
and radial acceleration it must have?