Ch 7 - Circular Motion

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Ch 7 - Circular Motion

• Circular motion Objects moving in a circular
  path.

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Measuring Rotational Motion

• Rotational Motion when an object turns about an
  internal axis.
• Ex. Earths is every 24 hrs.
• Axis of rotation the line about which the
  rotation occurs
• Arc length the distance (s) measured along the
  circumference of the circle

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• Radian- an angle whose arc length is equal to its
  radius, which is approximately equal to 57.3
• When the arc length s is equal to the length of
  the radius, r, the angle ? swept by r is equal
  to one rad.
• Any angle ? is radians if defined by
• ? s/r

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• When a point moves 360,
• ?s/r 2pr/r 2p rad
• Therefore, to convert from rads to degrees
• ?(rad) p ?(deg)
• 180
• For angular displacement,
• ???s/r
• Angular displacement (in radians) change in arc
  length/distance from axis

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Example Problem

• While riding on a carousel that is rotating
  clockwise, a child travels through an arc length
  of 11.5 m. If the childs angular displacement
  is 165, what is the radius of the carousel?
• Ans. 3.98 m

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Angular Substitutes for Linear Quantities

• Linear (Straight Line)
• Displacement x
• Velocity v
• Acceleration a

• Rotational
• Displacement ?
• Velocity ?
• Acceleration a

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• Angular speed the rate at which a body rotates
  about an axis, expressed in radians per second
• Symbol ? (omega) Unit rad/s
• ?(avg) ??/?t
• ? can also be in rev/s
• To convert
• 1 rev 2p rad

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Example Problem

• A child at an ice cream parlor spins on a stool.
  The child turns counter-clockwise with an average
  angular speed of 4.0 rad/s. In what time
  interval will the childs feet have an angular
  displacement of 8.0 rad
• Ans. 6.3 s

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• Angular Acceleration the time rate of change of
  angular speed, expressed in radians per second
  per second
• avg ?2-?1/t2-t1 ??/?t
• Average angular acceleration change in angular
  speed / time interval

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Example Problem

• A cars tire rotates at an initial angular speed
  of 21.5 rad/s. The driver accelerates, and after
  3.5 s the tires angular speed is 28.0 rad/s.
  What is the tires average angular acceleration
  during the 3.5 s time interval?
• Ans. 1.9 rad/s2

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Frequency vs. Period

• Frequency of revolutions per unit of time.
  Unit revolutions/second (rev/s).
• Period time for one revolution. Unit second
  (s).
• Inversely related
• t 1/f and f 1/t

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Tangential Velocity

• Speed that moves along a circular path.
• Right angles to the radii.
• Direction of motion is always tangent to the
  circle.

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Rotational Speed

• The number of rotations per unit of time.
• All parts of the object rotate about their axis
  in the same amount of time.
• Units RPM (revolutions per minute).

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Tangential vs. Rotational

• If an object is rotating
• All points on the object have the same
  rotational (angular) velocity.
• All points on the object do not have the same
  linear (tangential) velocity.
• Tangential speed is greater on the outer edge
  than closer to the axis. A point on the outer
  edge moves a greater distance than a point at the
  center.
• Tangential speed radial distance x rotational
  speed

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Centripetal Acceleration

• The acceleration of an object moving in a circle
  points toward the center of the circle.
• Means center seeking or toward the center.

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7.3 Forces that maintain circular motion
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• Consider a ball swinging on a string. Inertia
  tends to make the ball stay in a straight-line
  path, but the string counteracts this by exerting
  a force on the ball that makes the ball follow a
  circular path.
• This force is directed along the length of the
  string toward the center of the circle.

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The force that maintains circular motion
(formerly known as centripetal force)

• Fc (mvt2)/r
• Force that maintains circular motion mass x
  (tangential speed)2 distance to axis of motion
• Fc mr?2
• Force that maintains circular motion mass x
  distance to axis x (angular speed)2
• Because this is a Force, the SI unit is the
  Newton (N)

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Practice Problem

• A pilot is flying a small plane at 30.0 m/s in a
  circular path with a radius of 100.0 m. If a
  force of 635N is needed to maintain the pilots
  circular motion, what is the pilots mass?
• Answer m 70.6 kg

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Common Misconceptions

• Inertia is often misinterpreted as a force
• Think of this example How does a washing
  machine remove excess water from clothes during
  the spin cycle?

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Newtons Law of Universal Gravitation

• Gravitational force a field force that always
  exists between two masses, regardless of the
  medium that separates them the mutual force of
  attraction between particles of matter
• Gravitational force depends on the distance
  between two masses

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Newtons Law of Universal Gravitation
Where F Force M1 and m2 are the masses of the
two objects R is the distance between the
objects And G 6.673 x 10-11 Nm2/kg2 (constant
of universal gravitation)
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Practice Problem

• Find the distance between a 0.300 kg billiard
  ball and a 0.400 kg billiard ball if the
  magnitude of the gravitational force is 8.92 x
  10-11 N.
• Answer r 3.00 x 10-1 m