Phys110_Lecture18Rotational Kinematics

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					Physic 110          Lecture 18
 from Chapter 7 Sections 1 to 3




    Rotational Kinematics
 Homework Assignment 18:

Problems:
  Chapter 7,   Problem   2 on page 218
  Chapter 7,   Problem   4 on page 219
  Chapter 7,   Problem   6 on page 219
  Chapter 7,   Problem   10 on page 219
The Radian
   The radian is a
    unit of angular
    measure
   The radian can be
    defined as the arc
    length s along a
    circle divided by
    the radius r
        s
    
        r
More About Radians
   Comparing degrees and radians

      1 revolution  360  2 rad
   Converting from degrees to
    radians
                
      [rad]        [deg rees]
               180
Example 1:
A bicycle with 68 cm diameter tires
 travels 2.0 km. How many
 revolutions do the wheels make?
 Example 1:
 A bicycle with 68 cm diameter tires
  travels 2.0 km. How many
  revolutions do the wheels make?
 r =34 cm = 0.34 m
 x = s = 2 km = 2000 m

   s 2000 m                1 rev
           5880 rad          936 rev
   r  0.34 m              2 rad
Angular Displacement
   Axis of rotation is
    the center of the
    disk
   Need a fixed
    reference line
   During time t, the
    reference line
    moves through
    angle θ
Angular Displacement:
 the angle the object rotates through

            f   i

where
  i    is the initial angular position

   f is the final angular position
Average Angular Speed
    The average
     angular speed, ω,
     of a rotating rigid
     object is the ratio
     of the angular
     displacement to
     the time interval
            f  i    
     av            
             tf  ti   t
Angular Speed, cont.
   The instantaneous angular speed is
    defined as the limit of the average
    speed as the time interval approaches
    zero
   Units of angular speed are radians/sec
       rad/s
   Speed will be positive if θ is increasing
    (counterclockwise)
   Speed will be negative if θ is decreasing
    (clockwise)
Example 2
Find the angular speed of the Earth
  around the sun in radians per
  second and degrees per day.
    Example 2
    Earth makes 1 rev around sun in 1 year.

      1rev 360 deg     1year
                              0.986o / day
   t 1year   1rev     365.25days
         deg   2 rad   1day   1hr
  0.986                         2x107 rad / s
         day 360 deg 24hr 3600s
Average Angular Acceleration
   the ratio of the change in the angular
    speed to the time it takes

             f  i        
     av                 
              tf  ti       t
Units of angular acceleration: rad/s2
 Analogies Between Linear
 and Rotational Motion
  Uniform motion: ω = constant

 Linear:                     Rotational:

 x f  xi  v t            f  o   t
where θi and θf are the angular displacements
      and ω is the angular velocity
   Analogies Between Linear
   and Rotational Motion
   Uniform accelerated motion: α= constant

   Linear:                     Rotational:
v f  vo  a t               f  o   t

x f  xo   vo  v f  t    f   o   o   f  t
           1                            1
           2                            2
                  1 2                         1 2
x f  xo  vot  a t         f   o  o t   t
                  2                           2
v 2  vo  2a( x f  xo )
  f
       2
                             2  o2  2 ( f   o )
                               f
  Example 3:
A tire on a balancing machine starts from rest
  and turns through 4.7 revolutions in 1.2 s
  before reaching its final speed.

Assuming the acceleration was constant, find
a) the angular velocity at the end of this time
b) the angular acceleration
     Example 3:
  θo = 0.0              θf = 4.7 rev = 29.5 rad
  t = 1.2 s             ωo = 0.0 rad/s
  ωf = ?                 α=?
  For uniform angular acceleration:
    f   o  o   f  t
               1
               2
                 2  f   o 
    f  o 
                       t
          2  29.5rad  0 
f  0                         49.2rad / s
                1.2s
  Example 3:
θo = 0.0          θf = 4.7 rev = 29.5 rad
t = 1.2 s         ωo = 0.0 rad/s
ωf =49.2 rad/s     α=?
For uniform angular acceleration:
  f  o   t
        f  o
  
           t
   49.2rad / s  0
                  41rad / s 2

       1.2s
      Relationship Between Angular
      and Linear Quantities
    Consider a ball moving            arc length
                                        s r

    along an arc. v at
                                    velocity along arc
                       an               v  r
               r             s
                                   acceleration along arc
          ω
                   θ
                                         at   r
          α
                                 acceleration normal to arc
                                                    2
                                              v
                                   an   r 
                                            2

                                               r
   Example 4:
A 7.60 m diameter helicopter rotor rotates at a
  constant speed of 450 rev/min.
What is the speed of the tip?
What is the acceleration of the tip?

r = 7.6/2 = 3.8 m    ω = 450 rev/min = 47.1 rad/s

   v  r
       47.1rad / s  3.8m  179 m / s
   Example 4:
A 7.60 m diameter helicopter rotor rotates at a
  constant speed of 450 rev/min.
What is the speed of the tip?
What is the acceleration of the tip? (α = 0 rad/s2)

at   r                     an   r   2


  0rad / s 2  3.8m             47.1rad / s  (3.8m)   2


   0m / s   2
                                680m / s   2

				
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