Rotational Motion

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Rotational Motion

• Rotation of rigid objects- object with definite
  shape

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A brief lesson in Greek

• ? theta
• ? tau
• ? omega
• ? alpha

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Rotational Motion

• All points on object move in circles
• Center of these circles is a lineaxis of
  rotation
• What are some examples of rotational motion?

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Radians

• Angular position of object in degreesø
• More useful is radians
• 1 Radian angle subtended by arc whose length
  radius

Øl/r
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Converting to Radians

• If lr then ?1rad
• Complete circle 360º soin a full circle
  360??l/r2pr/r2prad
• So 1 rad360?/2p57.3?
• CONVERSIONS 1rad57.3?
• 360?2prad

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Example A ferris wheel rotates 5.5 revolutions.
How many radians has it rotated?

• 1 rev360?2prad6.28rad
• 5.5rev(5.5rev)(2prad/rev)
• 34.5rad

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Example Earth makes 1 complete revolution (or
2?rad) in a day. Through what angle does earth
rotate in 6hours?

• 6 hours is 1/4 of a day
• ?2?rad/4?rad/2

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Practice

• What is the angular displacement of each of the
  following hands of a clock in 1hr?
• Second hand
• Minute hand
• Hour hand

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Hands of a Clock

• Second -377rad
• Minute -6.28rad
• Hour -0.524rad

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Velocity and Acceleration

• Velocity is tangential to circle- in direction of
  motion
• Acceleration is towards center and axis of
  rotation

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Angular Velocity

• Angular velocity ? rate of change of angular
  position
• As object rotates its angular displacement is
  ???2-?1
• So angular velocity is
• ? ??/ ?t measured in rad/sec

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Angular Velocity

• All points in rigid object rotate with same
  angular velocity (move through same angle in same
  amount of time)
• Direction
• clockwise is -
• counterclockwise is

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VelocityLinear vs Angular

• Each point on rotating object also has linear
  velocity and acceleration
• Direction of linear velocity is tangent to circle
  at that point
• the hammer throw

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VelocityLinear vs Angular

• Even though angular velocity is same for any
  point, linear velocity depends on how far away
  from axis of rotation
• Think of a merry-go-round

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VelocityLinear vs Angular

• v ?l/?tr??/?t
• vr?

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Angular Acceleration

• If angular velocity is changing, object would
  undergo angular acceleration
• ? angular acceleration
• ???/?t
• Rad/s2
• Since ? is same for all points on rotating
  object, so is ? so radius does not matter

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Angular and Linear Acceleration

• Linear acceleration has 2 components tangential
  and centripetal
• Total acceleration is vector sum of 2 components
• aatangentialacentripetal

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Linear and Angular Measures
Quantity Linear Angular Relationship
Displacement d(m)
Velocity v(m/s)
Acceleration a(m/s2)
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Linear and Angular Measures
Quantity Linear Angular Relationship
Displacement d(m) ?(rad) dr ?
Velocity v(m/s) ?(rad/s) vr ?
Acceleration a(m/s2) ?(rad/s2) ar ?
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Practice

• If a truck has a linear acceleration of 1.85m/s2
  and the wheels have an angular acceleration of
  5.23rad/s2, what is the diameter of the trucks
  wheels?

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Truck

• Diameter0.707m

• Now say the truck is towing a trailer with wheels
  that have a diameter of 46cm
• How does linear acceleration of trailer compare
  with that of the truck?
• How does angular acceleration of trailer wheels
  compare with the truck wheels?

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Truck

• Linear acceleration is the same
• Angular acceleration is increased because the
  radius of the wheel is smaller

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Frequency

• Frequency f revolutions per second (Hz)
• PeriodTtime to make one complete revolution
• T 1/f

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Frequency and Period example

• After closing a deal with a client, Kent leans
  back in his swivel chair and spins around with a
  frequency of 0.5Hz. What is Kents period of
  spin?

T1/f1/0.5Hz2s
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Period and Frequency relate to linear and angular
acceleration

• Angle of 1 revolution2?rad
• Related to angular velocity
• ?2?f
• Since one revolution 2?r and the time it takes
  for one revolution T
• Then v 2?r /T

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Try it

• Joes favorite ride at the 50th State Fair is the
  Rotor. The ride has a radius of 4.0m and takes
  2.0s to make one full revolution. What is Joes
  linear velocity on the ride?

V 2?r /T 2?(4.0m)/2.0s13m/s
Now put it together with centripetal
acceleration what is Joes centripetal
acceleration?
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And the answer is

• Av2/r(13m/s2)/4.0m42m/s2

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Centripetal Acceleration

• acceleration change in velocity (speed and
  direction) in circular motion you are always
  changing direction- acceleration is towards the
  axis of rotation
• The farther away you are from the axis of
  rotation, the greater the centripetal
  acceleration
• Demo- crack the whip
• http//www.glenbrook.k12.il.us/gbssci/phys/mmedia/
  circmot/ucm.gif

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Centripetal examples

• Wet towel
• Bucket of water
• Beware.inertia is often misinterpreted as a
  force.

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The f word

• When you turn quickly- say in a car or roller
  coaster- you experience that feeling of leaning
  outward
• Youve heard it described before as centrifugal
  force
• Arghhthe f word
• When you are in circular motion, the force is
  inward- towards the axis centripetal
• So why does it feel like you are pushed out???

INERTIA
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Centripetal acceleration and force

• Centripetal accelerationv2/r
• Towards axis of rotation
• Centripetal forcemacentripetal

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Rolling
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Rolling

• Rolling rotation translation
• Static friction between rolling object and ground
  (point of contact is momentarily at rest so
  static)
• vr?

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Example p. 202

• A bike slows down uniformly from v8.40m/s to
  rest over a distance of 115m. Wheel diameter
  68.0cm. Determine
• angular velocity of wheels at t0
• total revolutions of each wheel before coming to
  rest
• angular acceleration of wheel
• time it took to stop

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Torque
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How do you make an object start to rotate?

• Pick an object in the room and list all the ways
  you can think of to make it start rotating.

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Torque

• Lets say we want to spin a can on the table. A
  force is required.
• One way to start rotation is to wind a string
  around outer edge of can and then pull.
• Where is the force acting?
• In which direction is the force acting?

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Torque
Force acting on outside of can. Where string
leaves the can, pulling tangent.
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Torque

• Where you apply the force is important.
• Think of trying to open a heavy door- if you push
  right next to the hinges (axis of rotation) it is
  very hard to move. If you push far from the
  hinges it is easier to move.
• Distance from axis of rotation
• lever arm or moment arm

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Torque

• Which string will open the door the easiest?
• In which direction do you need to pull the string
  to make the door open easiest?

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Torque
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Torque

• ? tau torque (mN)
• If force is perpendicular, ? rF
• If force is not perpendicular, need to find the
  perpendicular component of F
• ? rFsin?
• Where ? angle btwn F and r

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Torque example (perpendicular)

• Ned tightens a bolt in his car engine by exerting
  12N of force on his wrench at a distance of 0.40m
  from the fulcrum. How much torque must he
  produce to turn the bolt? (force is applied
  perpendicular to rotation)

Torque ? rF(12N)(0.4m)4.8mN
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Torque- Example glencoe p. 202

• A bolt on a car engine needs to be tightened with
  a torque of 35 mN. You use a 25cm long wrench
  and pull on the end of the wrench at an angle of
  60.0? from perpendicular. How long is the lever
  arm and how much force do you have to exert?
• Sketch the problem before solving

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More than one Torque

• When ?1 torque acting, angular acceleration ? is
  proportional to net torque
• If forces acting to rotate object in same
  direction net torquesum of torques
• If forces acting to rotate object in opposite
  directions net torquedifference of torques
• Counterclockwise
• Clockwise -

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Multiple Torque experiment

• Tape a penny to each side of your pencil and then
  balance pencil on your finger.
• Each penny exerts a torque that is equal to its
  weight (force of gravity) times the distance r
  from the balance point on your finger.
• Torques are equal but opposite in direction so
  net torque0
• If you placed 2 pennies on one side, where could
  you place the single penny on the other side to
  balance the torques?

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Torque and center of mass

• Stand with your heels against the wall and try to
  touch your toes.
• If there is no base of support under your center
  of mass you will topple over

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Torque and football

• If you kick the ball at the center of mass it
  will not spin
• If you kick the ball above or below the center of
  mass it will spin

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Inertia

• Remember our friend, Newton?
• Fma
• In circular motion
• torque takes the place of force
• Angular acceleration takes the place of
  acceleration

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Rotational InertiaLAZINESS

• Moment of inertia for a point object
• I Resistance to rotation
• Imr2 ?? I ?
• I plays the same role for rotational motion as
  mass does for translational motion
• I depends on distribution of mass with respect to
  axis of rotation
• When mass is concentrated close to axis of
  rotation, I is lower so easier to start and stop
  rotation

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Rotational InertiaUnlike translational motion,
distribution of mass is important in rotational
motion.
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Changing rotational inertia

• When you change your rotational inertia you can
  drastically change your velocity
• So what about conservation of momentum?

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Angular momentum

• Momentum is conserved when no outside forces are
  acting
• In rotation- this means if no outside torques are
  acting
• A spinning ice skater pulls in her arms
  (decreasing her radius of spin) and spins faster
  yet her momentum is conserved

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Angular momentum

• Angular momentumLmvr
• Unit is kgm2/s

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Examples

• Hickory Dickory Dock
• A 20.0g mouse ran up a clock and took turns
  riding the second hand (0.20m), minute hand
  (0.20m), and the hour hand (0.10m). What was the
  angular momentum of the mouse on each of the 3
  hands?
• Try as a group.