8.1 Circular Motion

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8.1 Circular Motion
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Chapter 8 Objectives

• Calculate angular speed in radians per second.
• Calculate linear speed from angular speed and
  vice-versa.
• Describe and calculate centripetal forces and
  accelerations.
• Describe the relationship between the force of
  gravity and the masses and distance between
  objects.
• Calculate the force of gravity when given masses
  and distance between two objects.
• Describe why satellites remain in orbit around a
  planet.

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Chapter 8 Vocabulary

• linear speed
• orbit
• radian
• revolve
• rotate
• satellite

• angular displacement
• angular speed
• axis
• centrifugal force
• centripetal acceleration
• centripetal force
• circumference
• ellipse
• gravitational constant
• law of universal gravitation

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Inv 8.1 Motion in Circles

• Investigation Key Question
• How do we describe circular motion?

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8.1 Motion in Circles

• We say an object rotates about its axis when the
  axis is part of the moving object.
• A child revolves on a merry-go-round because he
  is external to the merry-go-round's axis.

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8.1 Motion in Circles

• Earth revolves around the Sun once each year
  while it rotates around its north-south axis once
  each day.

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8.1 Motion in Circles

• Angular speed is the rate at which an object
  rotates or revolves.
• There are two ways to measure angular speed
• number of turns per unit of time
  (rotations/minute)
• change in angle per unit of time (deg/sec or
  rad/sec)

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8.1 Circular Motion

• A wheel rolling along the ground has both a
  linear speed and an angular speed.

• A point at the edge of a wheel moves one
  circumference in each turn of the circle.

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8.1 The relationship between linear and angular
speed

• The circumference is the distance around a
  circle.
• The circumference depends on the radius of the
  circle.

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8.1 The relationship between linear and angular
speed

• The linear speed (v) of a point at the edge of a
  turning circle is the circumference divided by
  the time it takes to make one full turn.
• The linear speed of a point on a wheel depends on
  the radius, r, which is the distance from the
  center of rotation.

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8.1 The relationship between linear and angular
speed
Radius (m)
C 2p r
Circumference (m)
Distance (m)
2p r
v d t
Speed (m/sec)
Time (sec)
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8.1 The relationship between linear and angular
speed
Radius (m)
v w r
Linear speed (m/sec)
Angular speed (rad/sec)
Angular speed is represented with a lowercase
Greek omega (?).
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Calculate linear from angular speed
Two children are spinning around on a
merry-go-round. Siv is standing 4 meters from
the axis of rotation and Holly is standing 2
meters from the axis. Calculate each childs
linear speed when the angular speed of the merry
go-round is 1 rad/sec?

• You are asked for the childrens linear speeds.
• You are given the angular speed of the
  merry-go-round and radius to each child.
• Use v ?r
• Solve
• For Siv v (1 rad/s)(4 m) v 4 m/s.
• For Holly v (1 rad/s)(2 m) v 2 m/s.

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8.1 The units of radians per second

• One radian is the angle you get when you rotate
  the radius of a circle a distance on the
  circumference equal to the length of the radius.
• One radian is approximately 57.3 degrees, so a
  radian is a larger unit of angle measure than a
  degree.

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8.1 The units of radians per second

• Angular speed naturally comes out in units of
  radians per second.
• For the purpose of angular speed, the radian is a
  better unit for angles.

• Radians are better for angular speed because a
  radian is a ratio of two lengths.

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8.1 Angular Speed
Angle turned (rad)
w q t
Angular speed (rad/sec)
Time taken (sec)
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Calculating angular speedin rad/s
A bicycle wheel makes six turns in 2 seconds.
What is its angular speed in radians per second?

1. You are asked for the angular speed.
2. You are given turns and time.
3. There are 2p radians in one full turn. Use ?
   ? t
4. Solve ? (6 2p) (2 s) 18.8 rad/s

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8.1 Relating angular speed, linear speed
and displacement

• As a wheel rotates, the point touching the ground
  passes around its circumference.
• When the wheel has turned one full rotation, it
  has moved forward a distance equal to its
  circumference.
• Therefore, the linear speed of a wheel is its
  angular speed multiplied by its radius.

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Calculating angular speedfrom linear speed
A bicycle has wheels that are 70 cm in diameter
(35 cm radius). The bicycle is moving forward
with a linear speed of 11 m/s. Assume the bicycle
wheels are not slipping and calculate the angular
speed of the wheels in rpm.

• You are asked for the angular speed in rpm.
• You are given the linear speed and radius of the
  wheel.
• Use v ?r, 1 rotation 2p radians
• Solve ? v r (11 m/s) (0.35 m) 31.4
  rad/s.
• Convert to rpm 31.4 rad x 60 s x 1 rotation
  300 rpm
• 1 s 1 min 2 p rad