Chapter 7: Circular Motion and Gravitation

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Chapter 7 Circular Motion and Gravitation
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Rotation vs. Revolution

• Axis The point at which rotation or revolution
  takes place.
• Exampleobjects turn about an axis (my hand
  during the ball and string demo)
• Rotation When the axis of rotation lies within
  the body.
• The Earth Rotates once every 24 hours about its
  axis.
• Revolution When the axis of rotation lies
  outside of the body.
• The Earth revolves around the sun once every 365¼
  days.

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Angular Measure

• When we discuss speed and velocity we have always
  used Cartesian coordinates x and y to represent
  distances.
• For circular motion we will use Polar Coordinates
  to describe our distances.
• Polar Coordinates are r radius from the
  origin and ? angle measured counterclockwise
  from the x-axis.
• Transformation equations for polar to cartesian
  is
• x r cos ?
• y r sin ?

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• Angular Displacement Just like Linear
  displacement
• ?? ? ?o
• Arc Length (s) The distance traveled along the
  circular path. Measured in meters.
• Radian The unit used to relate angle to arc
  length. The angle defining an arc length (s)
  that is equal to the radius (r).
• 2p rad 360º
• s r ?, where ? is measured in rad.

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Angular Speed and Velocity

• Angular Speed is very similar to linear speed.
• Angular Speed (?) displacement / the total time
  to travel the distance.
• ? ??/?t, Units rad/s, RPM is also used
• Angular Velocity particles travel with an angular
  displacement either clockwise or
  counterclockwise.
• The direction of the actual angular velocity is
  found by using the Right Hand Rule.
• The fingers of your hand fold around the axis in
  the direction of motion. The way that your
  thumb points is the direction of your angular
  velocity vector.

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Tangential and Angular Speed

• Tangential Speed (?-greek letter nu) is the speed
  of an object tangent to the circle at which it is
  rotating/revolving.
• ? r ?, units are m/s

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Period and Frequency

• The Period (T) is the time it takes for an object
  in circular motion to make one complete
  revolution or cycle.
• Period of revolution for the Earth about the sun
  is 1 year.
• Period of the Earths axial rotation is 24 hours.
• T 1/f
• Frequency (f) is the number of revolutions or
  cycles made in a given time, generally a second.
• f 1/T, Units 1/s, which is called Hertz (Hz)
• Angular Speed ? 2p / T 2p f

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Uniform Circular Motion and Centripetal
Acceleration

• Uniform Circular Motion- Constant speed but not
  constant velocity.
• A car going around a circular track
• Centripetal Acceleration- center-seeking
  acceleration.
• ac ?2/r
• Can also be written in terms of angular speed
• ac ?2/r (r ?)2/r r ?2

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Centripetal Force

• Centripetal Force net inward force
• Fc mac m ?2/r
• Centripetal Force, like centripetal acceleration
  is directed toward the middle or center.

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Angular Acceleration

• Just like regular acceleration is the change in
  veolcity divided by time, Average Angular
  Acceleration is the change in angular velocity
  divided by time
• a ? ? / ? t ? ?0 / t
• ? ?0 a t

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Tangential Acceleration

• Associated with the tangential speed changes and
  hence continuously changes direction.
• at ?? / ?t ?(r ?) / ?t r ? ? / ?t r a

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Equations for Linear and Angular Motion with
Constant Acceleration. (P. 227)

• x v t
• v (v v0)/2
• v v0 at
• x v0t (1/2) at2
• v2 v02 2 ax

• ? ?t
• ? (? ?0)/2
• ? ?0 a t
• ? ?0t (1/2)a t2
• ? 2 ?02 2 a?

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Newtons Law of Gravitation

• Universal Law of Gravitation describes the
  relationship for the gravitational interaction
  between 2 particles, or point masses, m1 and m2,
  separated by a distance r.
• F (G m1 m2 )/ r2
• Where G is the universal gravitational constant
• G 6.67 x 10-11Nm2/kg2

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Finding acceleration due to gravity

• Using Newtons 2nd Law along with the Law of
  Gravitation, we can find the acceleration due to
  gravity of a mass at a distance r from the
  planets center.
• F (G m1 M2 )/ r2
• Where Mplanet mass, and m point mass.
• mag (G m1 M2 )/ r2
• ag (G M2 )/ r2

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Acceleration due to gravity does vary with
altitude

• By looking at the equation for acceleration due
  to gravity, we can see that we can calculate the
  radius from the center if we know the Mass and
  acceleration due to gravity. ag (G Me )/ Re2
• To find the height above the surface, we can
  simply look at the equation like this
• ag (G Me )/ (Re h)2

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Gravitational Potential Energy

• The Gravitational Potential Energy for 2 point
  masses separated by a distance r is given by
• U - (G m1 M2 )/ r (G Me )/ (Re h)

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Keplers Laws

• 1st Law (the law of orbits) Planets move in
  elliptical orbits with the Sun at one of the
  focal points.
• 2nd Law (the law of areas) A line from the Sun
  to a planet sweeps out equal areas in equal
  lengths of time.
• 3rd Law (the law of periods) The square of the
  orbital period of a planet is directly
  proportional to the cube of the average distance
  of the planet from the Sun, T2 is proportional to
  r 3

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Earth Satellites

• Escape Speed The initial speed needed to escape
  from the surface of a planet or a moon.
• KE PE
• (1/2) mv2esc (GmMe)/Re
• Since g (GMe)/Re, we can say that
• vesc ?(2gRe)
• This equation can be used for the escape speed of
  Earth. However the base equation above can be
  used to find th escape speeds for other planets
  and our moon.