Chapter 7: Circular Motion and Gravitation

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Chapter 7Circular Motion and Gravitation

• Coach Kelsoe
• Physics
• Pages 233267

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Section 71Circular Motion

• Coach Kelsoe
• Physics
• Pages 234239

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Section 71 Objectives

• Solve problems involving centripetal
  acceleration.
• Solve problems involving centripetal force.
• Explain how the apparent existence of an outward
  force in circular motion can be explained as
  inertia resisting the centripetal force.

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Centripetal Acceleration

• The cars on a rotating Ferris wheel are said to
  be in circular motion.
• Any object that revolves about a single axis
  undergoes circular motion.

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Tangential Speed

• The line about which the rotation occurs is
  called the axis of rotation.
• Tangential speed (vt) can be used to describe the
  speed of an object in circular motion.

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Tangential Speed

• The tangential speed of a car on the Ferris wheel
  is the cars speed along an imaginary line drawn
  tangent to the cars circular path. This
  definition can be applied to any object moving in
  circular motion.
• When the tangential speed is constant, the motion
  is described as uniform circular motion.

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Tangential Speed

• Tangential speed depends on the distance from the
  object to the center of the circular path.
• To understand this concept, imagine a carousel.
  The horses or carts on a carousel are staggered
  so that some are on the outside edge while some
  are closer to the middle.

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Tangential Speed

• Each horse/cart completes one circle in the same
  time period, but the outside ones cover more
  area, therefore must have a greater tangential
  speed.

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Centripetal Acceleration

• If the cart on a Ferris wheel is moving at a
  constant speed around the wheel, it still has an
  acceleration.
• Even though we typically think of acceleration
  being a change of speed, it can also be a change
  of direction.
• On a Ferris wheel, the carts are constantly
  changing direction.

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Centripetal Acceleration

• An acceleration of this nature is called a
  centripetal acceleration the acceleration
  directed toward the center of a circular path.
• The equation for finding the magnitude of
  centripetal acceleration is given below

ac vt2/r
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Centripetal Acceleration

• Since acceleration is a vector quantity, we need
  to know the direction of the acceleration. But
  if direction constantly changes, how can we
  accurately define the direction?
• Centripetal acceleration is always toward the
  center of the circle! The word centripetal
  actually means center seeking.
• We can better understand this idea by drawing
  tangent vector lines or by adding opposite
  vectors at two points.

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Sample Problem A

• A test car moves at a constant speed around a
  circular track. If the car is 48.2 m from the
  tracks center and has a centripetal acceleration
  of 8.05 m/s2, what is the cars tangential speed?

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Sample Problem A

• Given
• r 48.2 m
• ac 8.05 m/s2
• Unknown
• vt ?
• Solve
• ac vt2/r , so
• vt vacr v(8.05 m/s2)(48.2 m)
• vt 19.7 m/s

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Tangential Acceleration

• Centripetal acceleration results from a change in
  direction, not a change in speed.
• In circular motion, an acceleration due to a
  change in speed is called tangential
  acceleration.
• The easiest way to think of this is a car on a
  circular track it has centripetal acceleration
  no matter what, due to its change in direction.
  It will only have tangential acceleration if it
  speeds up or slows down.

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Centripetal Force

• Consider a ball of mass m that is tied to a
  string of a length r and that is being whirled in
  a horizontal circular path.
• Assume the ball moves with a constant speed.

r
m
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Centripetal Force

• Assume that the ball moves with constant speed.
  Because the velocity vector, v, continuously
  changes direction during the motion, the ball
  experiences a centripetal acceleration that is
  directed toward the center of the motion.
• The inertia of the ball tends to maintain the
  balls motion in a straight path. However, the
  string exerts a force that overcomes this
  tendency.

r
m
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Centripetal Force

• The net force that is directed toward the center
  of an objects circular path is called
  centripetal force.
• Newtons second law can be applied to find the
  magnitude of this force Fc mac.
• The equation for centripetal acceleration can be
  combined with Newtons second law to obtain the
  following equation
• Fc mvt2/r

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Centripetal Force
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Centripetal Force

• Centripetal force is simply the name given to the
  net force on an object in uniform circular
  motion. Any type of force or combination of
  forces can provide this net force.
• Example Friction between a race cars tires and
  a circular track is a centripetal force that
  keeps the car in a circular path.
• Example Gravitational force is a centripetal
  force that keeps the moon in its orbit.

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Centripetal Force

• Because centripetal force acts at right angles to
  an objects circular motion, the force changes
  the direction of the objects velocity. If this
  force vanishes, the object stops moving in a
  circular path and instead, moves along a straight
  line path that is tangent to the circle.

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Rotating Systems

• Think about the feelings you experience as you
  made a sharp turn in your vehicle. If you make a
  sharp turn to your right, you are thrown against
  the door to the left.
• If centripetal force is always toward the center
  of the circular path, why wouldnt you be thrown
  toward the inside of the car rather than the
  outside?
• A popular explanation is that a force must push
  you outward. Many times this force is called
  centrifugal force, but to lessen confusion, we
  will refrain from using this term.

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Rotating Systems

• So lets explain the false force that is
  centrifugal force.
• Before you begin to make your turn, your body is
  following a straight-line path. As the car
  enters the turn, your inertia makes to tend to
  move along the original straight-line path. This
  movement is in accordance with Newtons first
  law, which states that the natural tendency of a
  body is to continue moving in a straight line.

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Rotating Systems

• If a large centripetal force acts on you, you
  will move in the same direction as the car. The
  origin of this force is the force of friction
  between you and the car seat.
• Think about it if your seat was slippery, and
  the door wasnt there, youd slide right out!
• This gives you a great reason not to Armor-All
  your seats in a Jeep with no doors!

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Inertia, not Centrifugal Force!
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Simulated Gravity Using Centripetal Force!

• http//www.courses.psu.edu/aersp/aersp055_r81/stat
  ion/station.html

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