Circular Motion

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Circular Motion

• Chapter 7.3

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What moves in a circle?

• The earth around the sun
• A car on a curve
• A disk on a string
• A tetherball

Day 1
3
Why does it move in a circle?

• The earth around the sun
• Gravity
• A car on a curve
• Friction
• A disk on a string
• The tension in the string
• A tetherball
• The tension in the string

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Centripetal Force

• We call any force that keeps something moving in
  a circle a centripetal force
• In order to keep something moving in a circle, we
  must provide a force to counter the tendency of
  inertia
• What does the inertia of the object make it want
  to do?

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Centripetal Acceleration

• To keep something moving in a circle we must
  constantly provide a force
• The direction of the velocity is constantly
  changing
• What do we call a change in velocity (a vector
  quantity) with time?
• Is a body moving in a circle at equilibrium?
• Is there a net unbalanced force acting on the
  object? (Is there an acceleration?)

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Centripetal Force

• Centripetal force is affected by
• The mass of the object (m).
• The speed of the object around the circle (v).
• The radius of the circle (r).
• Using Newtons 2nd Law of Motion (F ma),
  centripetal force is mathematically represented
  as follows
• Fc mac mv2 ac v2
  
  r r

Day 1
7
Motion Forces

• What you already know
• Velocity a measure of the change in
  displacement (distance with direction.
• Mass A measure of the amount of matter an
  object contains.
• Acceleration A measure of the change in
  velocity over change in time.
• Force A push or pull that is equal to the mass
  of the object multiplied by its acceleration (F
  ma).

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Uniform Circular Motion

• Uniform circular motion is defined by any object
  that is moving at constant speed in a circular
  path.
• Determining Speed
• The distance an object moving in a circular path
  is equal to the circumference (C 2?r).
• The time it takes an object to complete one
  revolution is called the period (T).
• It then follows that the speed of an object
  moving in a circular path can be determined by
• v d/t C/T 2?r/T

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Uniform Circular Motion

• If an object is moving at constant speed in a
  circular path, can it be accelerating?
• Yes
• Although the speed may be constant, the direction
  is changing.
• If direction is changing over time, then the
  velocity must be changing.
• Acceleration is the change in velocity over time
  (a ?v/?t).
• If the velocity is changing over time, then the
  object must be accelerating.

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Circular Motion Instantaneous Velocity

• Note that the velocity vector is at right angles
  to the position vector and tangent to the circle
  at any given point along the circle.

v ?r/?t
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Circular Motion Centripetal Acceleration (ac)

• The acceleration of an object moving in a
  circular path always points towards the center of
  the circle, and is perpendicular to the velocity
  vector.

v2
? v
a
v1
r
a ?v/? t
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Centripetal Acceleration

• The angle between r1 and r2 is the same as the
  angle between v1 and v2.
• Therefore, the triangles these vectors make are
  similar such that
• ?r/r ?v/v
• If you divide both sides by ?t
• ?r/(?t r) ?v/(?tv)
• Since
• ?r/?t v and ?v/?t a
• Hence
• v/r a/v
• and
• ac v2/r

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Centripetal Acceleration

• An alternative representation for centripetal
  acceleration can be derived using the
  circumference and period of revolution.
• d 2pr
• v d/T 2pr/T
• Substituting into ac v2/r
• ac (2pr/T)2/r
• ac 4p2r/T2

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Circular Motion Centripetal Force

• To make an object move in a circular path, an
  external force must act perpendicular or at right
  angles to its direction of motion.
• This force is called centripetal force.

Instantaneous direction of velocity
Direction of force required to make object move
in a circular path (towards the center)
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Centripetal Force

• Centripetal force is affected by
• The mass of the object (m).
• The speed of the object around the circle (v).
• The radius of the circle (r).
• Using Newtons 2nd Law of Motion (F ma),
  centripetal force is mathematically represented
  as follows
• F mv2
• r

Note Centripetal force is an unbalanced net
force
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How the Factors Affect Centripetal Motion

• Which graph shows the proper relationship with
  respect to force
• Force vs. Mass.
• Force vs. Speed.
• Force vs. Radius.

Speed
Radius
Mass
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Objects that travel in circular paths. What is
the cause of the force?

• The Earth Sun System
• Gravity.
• A racecar traveling around a turn on the
  racetrack
• Friction.
• An athlete throwing the hammer
• Tension in the cable attached to the hammer.

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The path of objects.

• If the centripetal force were suddenly removed
  from an object moving in a circular path, what
  trajectory (or path) would it follow?

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Which Path?

• Why?
• Because of Inertia An object in motion wants
  to remain in motion at constant speed in a
  straight line.
• If the unbalanced centripetal force is removed,
  the object will continue in a straight path.

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Example 1

• A 1.5 kg cart moves in a circular path of 1.3
  meter radius at a constant speed of 2.0 m/s.
• Determine the magnitude of the centripetal
  acceleration. (ac v2/r )
• Determine the magnitude of the centripetal force.
  (Fc mac )
• Determine the period. (T d/v c/v)
• Get out your calculators!

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Example 1 (cont.)

• Centripetal Acceleration
• ac v2/r (2.0 m/s)2/(1.3 m) 3.1 m/s2
• Centripetal Force
• Fc mac (1.5 kg)(3.1 m/s2) 4.6 N
• Period
• T c/v 2?r/v 2?(1.3 m)/(2.0 m/s) 4.08 s

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Example 2 Roller Coaster
The centripetal force is the force keeping the
object in a circle. It is either going to be Fg,
FN, or some combination.
m 1kg r 10m v 9.9m/s ac ??m/s2 Fc ??N
Here Fg Fc. Fg IS Fc. There is no Normal
force. You feel weightless.
Fg mg Fc
Fc N
Fg mg
Fg mg
Here Fg Fc but point in opposite directions.
The Normal force has to oppose Fg and provide Fc.
You feel 2 gs.
Here Fc FN and points to the center. Fg points
straight down but if you are accelerating with it
you only feel Fc. (And feel confused if you have
time.)
Fg mg
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Example 2 Roller Coaster
The track of the roller coaster provides the
centripetal force via the Normal force, FN
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Example 3 Ball on a rope
A tetherball of mass 0.28kg is moving at 8m/sec
around a pole. (For this example, we will ignore
gravity.) The length of the rope is 2.4m.
What is the period of motion? What is the
centripetal force? Ignoring gravity, what is
the tension in the rope? (For this example, we
will ignore gravity.)
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Example 3 Ball on a rope
Period T d/v 2Pr/v 2P(2.4m)/(8m/s)
1.9s (This is not actually needed to solve the
problem.) What is the centripetal force? Fc
mac mv2/r 0.28kg (4m/s) 2/2.4m What is
the tension in the rope? The tension Fc
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Example 4 Car on a curve
A 1000kg car is traveling at 22m/s on a curve of
radius 100m. What provides the centripetal force
required to keep the car from sliding off the
road? What is the magnitude of the centripetal
force required to keep the car on the road? What
is the coefficient of friction between the car
and the road? Fc mv2/r 1000kg (22m/s)2/100m
4840N. What is the coefficient of friction? U
Ff/Fn
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Example 4 Car on a curve
What provides the centripetal force required to
keep the car from sliding off the road?
Friction What is the magnitude of the
centripetal force required to keep the car on the
road? Fc mv2/r 1000kg (22m/s)2/100m
4840N. What is the coefficient of friction
between the car and the road? m Ff/FN Fc/FN
Fc/(mg) 4840N/(1000kg 9.81m/s2) 0.5 What
is the coefficient of friction? U Ff/Fn
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Example 5 Moon around the Earth
The moon has a mass of 7.35 1022 kg and orbits
at an average distance of 3.84 108 m in 27.32
days (2.36 106sec). What centripetal
acceleration does the moon experience and what is
the required centripetal force? ac
0.0027m/s2 Fc mac (7.35 1022
kg)(0.0027m/s2) 1.98 1020N