Rotational Dynamics

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Rotational Dynamics
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Radian measure compared to degree measure

• 1 radian angle of rotation where the arc length
  of rotation radius of the circle

1 rad about 57 2p rad 360
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Rotational displacement

• Change in the angle of rotation
• Symbol is ? (theta), unit is rads
• Counterclockwise motion is
• Clockwise motion is -

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Rotational velocity

• Change in angular displacement over time
• Measures rate of angular motion
• ? ?? / t
• Symbol is ? (omega), unit is rads / sec

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Rotational Acceleration

• Change in the rate of angular velocity over time
• a ?? / t
• Symbol is a (alpha), unit is rads / sec2

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Relationship between linear and rotational motion
values

• Linear values for a rotating object depends on
  its distance from the pivot point
• Relationships between linear and angular values
  depend on the radius
• d ?r v ?r a ar

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Angular frequency

• Cycles through rotation
• ƒ ? / 2p
• Remember v 2pr / T and v ?r?

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Question

• What is the angular displacement of each of the
  following hands of a clock in 1 hour?
• The second hand
• The hour hand

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Answers

1. -377 rad
2. -0.524 rad

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Next question

• The rotational velocity of a merry-go-round in
  increased from 1.5 rads/s to 3.5 rads/s over 9.5
  seconds.
• What is the rotational acceleration of the
  merry-go-round?

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Homework on pg 200

• Question 1-10

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DEFINITION OF TORQUE
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Rotating a door

• Where is the best place to apply force to open
  the door?
• How does the direction that force is applied
  relate to the pivot point?

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Torque and pipe wrench, extenders
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Big ideas for this section

• After this section you should be able to
• define what torque is
• identify examples of torque in real life
• identify lever arm and applications points
• calculate torque

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Torque

• Is a measure of how effectively a force causes
  rotation.
• Generated by an application of force on an object
  in a direction that does not go through the pivot
  point

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Result of Torque

• Application of Torque can result in object
  changing the rate of spin

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Torque

• Symbol is T (Tau), Units are Nm
• Not Joules (no displacement)
• Direction (clockwise-, counter clockwise )

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How to calculate the torque

• Need
• Size of force applied F
• Distance between application of force and
  pivot point (lever arm length r )
• Angle between force and lever arm (use the
  smaller of the two angles T)

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Torque equation

• T rFsinA,
• Where
• r means______________
• F means _____________
• A means ______________

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Torque is larger when

• A larger force is applied
• The length of the lever arm is increased
• The angle between force and arm is 90

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Questions on torque

• A bolt of a car engine needs to be tightened with
  a torque of 35 Nm. You use a 25 cm long wrench
  and pull on the end of the wrench at an angle of
  120from body of the wrench.
• How much force do you exert on the wrench?

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Homework

• 12-15 on pg 203

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Answers to questions on pg 200

• a -120p or -377 rad
• b -2p or -6.28 rad
• c) p/6 rad
• 2) Diameter 0.707 m
• 3) a) Linear acc is same as truck
• b) a 7.71 rad/s2
• 4) Angular velocity decreases
• revolution decreases

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Answers to pg 200 cont

• 5) a -p/3 or -1.05 rad/s
• b) -4p or -12.6 rad
• 6) a 2358720 sec T
• b 4.24x10-8 cycles/s or 2.66x10-6 rad/s
• c V .46 m/s on moon
• d v 463 m/s on earth (1000 times larger)

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Answers for 200

• 7) 3.8p or 12 radians
• 8) Yes to same angular displacement
• No, to traveling the same linear distance
• 9) a - 8.3 rad/s2
• 10) ? -0.0059 rad/s2

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See-saws and torque
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Net torque

• Sum of all torque exerted on an object
• (Not the sum of all forces)
• (Not all forces exert torque)
• Tnet 0 means that the clockwise torque is
  balanced by the counterclockwise torque

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Net torque question

• 2 kids, the 1st is 65 kg , 2nd is 45 kg want to
  balance on a 3 m long seesaw.
• If the 45 kg kid wants to be at the end of the
  see-saw on the left, where would you place the
  other kid?
• Can they remain balanced only if the seesaw is
  horizontal?

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Homework

• Pg 205 16-19

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Answers 16-19 on pg 205

• 16) About 1.49 meters from center
• 17) About 2.7 Nm in the counterclockwise (or )
  direction
• 18) About 0.056 kg
• 19) About .042 kg

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What happens to an object if Tnet ? 0?
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Torque and Angular Acceleration

• When a rigid object is subject to a net torque
  (?0), it undergoes an angular acceleration
• The angular acceleration is directly proportional
  to the net torque
• The relationship is analogous to ?F ma
• Newtons Second Law

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What characteristic of the object resists change
in its rotation ?
35

• Another name for the measure of resistance to
  change in motion is ..

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Moment of Inertia

• The angular acceleration is inversely
  proportional
• to the objects mass
• its position of mass in a rotating system
• This mass component is called the moment of
  inertia, I, of the object
• Inertia of rotation

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Moment of Inertia

• I mr2
• (if a single point mass)
• where
• I moment of inertia
• m mass
• r distance mass is from pivot
• SI units are kg m2

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Which moves done the ramp faster (greater a) ?
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Moment inertia depends on the shape and size of
mass

• The farther and larger the mass is from the
  pivot, the greater tits moment of inertia

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More About Moment of Inertia

• There is a major difference between moment of
  inertia and mass the moment of inertia depends
  on the quantity of matter and its distribution in
  the rigid object.
• The moment of inertia also depends upon the
  location of the axis of rotation

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Homework on Moment of inertia and rotational
acceleration

• Pg 208 21-24

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Answers to homework

• 21) Done in class
• 22) Hollow ball has greater moment of inertia,
  mass is farther away
• 23) A has greater moment of inertia (5 mr2
  compared to 2 mr2)
• 24) .02 kg m2 compared to 0.008 kg m2
• Challenge least A (0), D (5), C (6) , B (14) most

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Newtons Second Law for a Rotating Object

• The angular acceleration is directly proportional
  to the net torque
• The angular acceleration is inversely
  proportional to the moment of inertia of the
  object

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Remember

• Torque is the application of force, not force
• Moment of inertia is based on the mass shape and
  position from the pivot point

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Example problem

• A solid steel wheel has a mass of 15 kg and a
  diameter of 0.44m. It starts from rest. You want
  to make it rotate at 8.0 rev/s in 15 s.
• What torque must be applied to the wheel?
• If you apply the torque by wrapping a strap
  around the wheel, how much force should you exert
  on the strap?

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Moment of Inertia of a Uniform Ring

• Image the hoop is divided into a number of small
  segments, m1
• These segments are equidistant from the axis

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Other Moments of Inertia
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Homework

• Pg 210 25-35
• To be turned in on Thursday

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Rotational Kinetic Energy

• An object rotating about some axis with an
  angular speed, ?, has rotational kinetic energy
  ½I?2
• Energy concepts can be useful for simplifying the
  analysis of rotational motion

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Total Energy of a System

• Conservation of Mechanical Energy
• Remember, this is for conservative forces, no
  dissipative forces such as friction can be
  present

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Angular Momentum

• Similarly to the relationship between force and
  momentum in a linear system, we can show the
  relationship between torque and angular momentum
• Angular momentum is defined as
• L I ?
• and

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Angular Momentum, cont

• If the net torque is zero, the angular momentum
  remains constant
• Conservation of Linear Momentum states The
  angular momentum of a system is conserved when
  the net external torque acting on the systems is
  zero.
• That is, when

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Problem Solving Hints

• The same basic techniques that were used in
  linear motion can be applied to rotational
  motion.
• Analogies F becomes , m becomes I and a
  becomes , v becomes ? and x becomes ?

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More Problem Solving Hints

• Techniques for conservation of energy are the
  same as for linear systems, as long as you
  include the rotational kinetic energy
• Problems involving angular momentum are
  essentially the same technique as those with
  linear momentum
• The moment of inertia may change, leading to a
  change in angular momentum