Reminder

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Reminder

• Please take advantage of my office hours
• MWF 11-noon
• By appointment gblazey_at_nicadd.niu.edu
• Help room 2nd floor Faraday
• The University has Retention Programs that can
  help, let me know if you are interested.

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Unit 4 Circular Motion and Gravity

• Weve fully investigated linear motion and
  forces. But this is somewhat limited.
• Now its time to take a look at circular motion
  and apply Newtons Laws
• To do so well collect the various sections of
  the book on circular motion.
• Beginning with uniform circular motion, angular
  variables, and the equations of motion for
  angular motion (3-9, 10-1, 10-2)
• Then well move onto
• Applications of Newtons Laws (5-2, 5-3, 5-4)
• Harmonic Motion (14-1, 14-2)
• The Universal Law of Gravitation, Satellites, and
  Keplers Laws (Chapter 6)

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Uniform Circular Motion (Section 3-9)

• Definition Motion of an object moving in a
  circle at constant speed.
• Examples
• Ball on a string
• Satellite in orbit around the earth.
• Stars around a galaxys center
• (a black hole?)
• Characteristics
• Magnitude of velocity constant
• Direction under continuous change
• Consequently there is an acceleration

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The Direction of the Acceleration of Circular
Motion

• Consider the figure at right
• During a small time interval Dt, a particle moves
  from A to B
• It covers a small arc labeled Dl and subtends a
  small angle Dq.
• The change in the velocity vector is just given
  by v2-v1Dv

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• Now consider taking Dt to the limit.
• As it becomes very small then Dl and Dq become
  very small.
• Also the two vectors v2 and v1 become almost
  parallel. Accordingly Dv will be perpendicular to
  both.
• This can only occur if Dv points toward the
  center of the circle.
• Recall that
• Which means a points toward the center as well
• Accordingly its called centripetal or radial
  acceleration aR

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What about the magnitude?

• The triangle ABC is congruent or geometrically
  similar to the triangle abc.
• Thus
• As shown w hen Dt approaches zero, the arc equals
  the chord.
• In that limit

a
c
b
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• But since
• then
• gives,

• The basic result is then
• an entirely geometric result.
• In words An object moving in a circle of radius
  r with constant speed v has an acceleration
  toward the center of the circle with magnitude
  aRv2/r.

aRv2/r
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Other Characteristics of Circular Motion

• Acceleration and velocity are perpendicular
• Frequency, f, is the number of revolutions per
  second
• Period, T, time required for one revolution.
• T 1/f
• v2pr/T

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The Moons Radial Acceleration.

• The moon orbits the earth at a radius of
  384,000km (3.84x108m) and with a period of 27.3
  days(2.36x106s). What is the centripetal
  acceleration?
• The circumference of the circle describing the
  orbit is 2pr.
• The velocity would be that circumference divided
  by the period T or 2pr/T.

• So aRv2/r (2pr/T)2/r 4p2r/T2
• We can just substitute our values for r and T
• aR4p2r/T2
• 4(3.14)2(3.84x108m)
  (2.36x106s)2
• 2.72x10-3 m/s2
• Presumably this is due to gravity? Compare to
  9.80m/s2

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Angular Quantities (Section 10-1)

• As we will see, there is a close parallelism
  between the variables of linear motion and those
  of angular motion.
• The motion of a rigid body can be described with
  both translation motion and rotational motion.
• Consider the disk at the right undergoing purely
  rotational motion, that is all points move in a
  circle about the axis of rotation, which is
  projecting from the screen.

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We use R rather than r to indicate distance from
the axis of rotation.
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Indicating Angular Position of a Point

• Given by an angle with respect to an axis.
• A point P moves through angle q as it travels
  along arc l.
• Angles can be given in degrees or more
  conveniently in radians.
• One radian is the angle subtended by an arc equal
  to the radius.
• Note that one radian is the same angle for any
  sized circle.

1 rad
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More on Radians

• By definition then
• q l/R
• where R is the radius of a circle,
• and l is the arc length subtended by q.
• Note radians are dimensionless!
• Radians are easily related to degrees since the
  360o arc length of a complete circle is 2pR
• 360o l/R 2pR/R 2p rads ? 1 rad 57.3o

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Angular Variables of Motion w and a

• Consider the wheel, its angular displacement
  after a bit of rotation is given by
• In complete analogy with average velocity the
  average angular velocity, w, is defined as
• And the instantaneous angular velocity is

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• We can also define average and instantaneous
  angular acceleration in analogy to linear
  acceleration
• The units of w are rad/s and for a they are
  rad/s2.

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The Relationship between Angular and Linear
Velocity

• Each point on a rotating rigid body has nonzero w
  and v.
• The figure helps to under-stand the relationship
  between the two for P.
• The magnitude of the linear velocity is given by

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vRw
Note that different radii have equal angular
velocity but very different linear velocity
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The Relationship between Angular and Linear
Acceleration

• If an objects angular velocity changes there
  will also be angular acceleration.
• Every point on the object will then undergo
  tangential acceleration.
• But also recall there is a radial acceleration

The greater R the greater the acceleration
think Crack the Whip
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Collecting Results

• We can also write the angular velocity, w, in
  terms of the frequency, f.
• Since
• A frequency of 1 rev/sec an angular velocity of
  2p rads/sec, we can say
• f w/2p or w2pf
• The unit of frequency rev/s is given the name
  hertz(Hz) and since revolutions are not a true
  unit (just a place keeper) 1Hz1s-1.

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• Heinrich Rudolf Hertz (1857 - 1894) a German
  physicist and mechanician for whom the hertz, an
  SI unit, is named. In 1888, he was the first to
  satisfactorily demonstrate the existence of
  electromagnetic radiation by building an
  apparatus to produce UHF radio waves (300 MHz and
  3 GHz) .

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An Example Parameters of a Hard Drive Rotating
at 5400 rpm.

• What is angular velocity?
• Speed _at_3.0cm from axis?
• Linear acc. at 3.0 cm.?

• How many 5.0mm bits can be written per second at
  3.0cm?
• If the disk takes 3.6 s to reach speed what is
  the average acceleration?

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Equations of Motion for Rotational Motion

• The definitions of average and instantaneous
  angular velocity and angular acceleration are
  identical to linear velocity and acceleration
  except for a variable change
• Recall the definitions of average and
  instantaneous velocity and acceleration led to
  the four equations of linear motion for constant
  acceleration.
• An identical analysis for angular motion at
  constant angular acceleration would lead to the
  same four equations with the replacement

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Note since the equations are identical there is
no need for a re-derivation, this is a pretty
common technique!
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An example Back to the Hard-drive

• How many revolutions did the hard-drive execute
  when accelerating from 0 to 5400 rpm in 3.6s?
• Well we know w00, w570 rad/s, a160rad/s2
• Were really after the total angle turned during
  this interval. Use the 2nd equation

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Solving Rotational Motion Problems

1. Draw the situation, showing direction of
   rotation.
2. Decide on positive and negative directions of
   motion.
3. Write down list of rotational kinematic
   variables, q, a, w, wo, and t.
4. Verify that 3 of 5 variables are known, then
   select appropriate equation.
5. If not enough information see if segments share
   information or look for constraints
6. If two solutions, exist choose the physical one.

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Second Example A blender

• A blender on "puree" has blades spinning at
  angular velocity of 375 rad/s. The blades are
  accelerated with the "blend" selection. They
  reach their final angular velocity after the
  blades have rotated through 44.0 rad (seven
  revolutions). The angular acceleration has a
  constant value of 1740 rad/s2. Find the final
  angular velocity of the blades.
• Well the picture here is pretty simple. Looking
  down on the blender we see the blades rotating
  counterclockwise at an initial angular speed of
  375 rad/s.

• A counterclockwise acceleration kicks in at 1740
  rad/s2.
• After a counterclockwise displacement of 44.0
  rad the final angular velocity is reached.
• Here Ive painted a picture for you and retained
  counterclockwise as the positive direction (Steps
  1 and 2). Now lets write the list (Step 3)

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• We have three variables and see that the final
  angular velocity is given by the last equation
  (Step 4)
• w2 wo2 2aq
• Now its just plug-and-chug

• w2 (375 rad/s)2 2(1740 rad/s2)(44.0rad)
  2.97 x 105 rad2/s2
• Taking the root, w /-542 rad/s.
• Since all the motion is in the positive direction
  the answer must also be positive (Step 6)
  w 542 rad/s.

Known Unknown
q 44 rad w?
a 1740 rad/s2 t?
w0375 rad/s
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Where we are and where we are going

• So now weve got equations of motion for
  rotational as well as linear kinematics!
• That means, just as we were with linear
  kinematics, we are now in a position to
  concentrate on the dynamics of uniform circular
  motion.
• That is, Friday well apply Newtons 2nd laws to
  circular motion.