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Rotational Kinematics

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					Rotational Kinematics
           Angular Position q




Degrees and revolutions:
            Angular Position q

Arc length s,
measured in radians:


        s
     q
        r

    q (radians)  q (degrees)  /180  q (rev) 2
Angular Velocity w
Sign of w
A small sphere attached to a light rigid rod rotates about an axis
perpendicular to and fixed to the other end of the rod. Relative to
the positive direction of the axis of rotation, the angular positions
of the sphere are negative, its angular velocity is negative, and its
angular acceleration is negative. The sphere is


1.   rotating clockwise and
     slowing down.                        71%
2.   rotating counterclockwise
     and slowing down.
3.   rotating clockwise and
     speeding up.
4.   rotating counterclockwise
     and speeding up.
5.   first rotating
     counterclockwise and then                  10%   10%         10%
     clockwise.
                                                             0%

                                           1     2      3     4         5
A small sphere attached to a light rigid rod rotates about an axis
perpendicular to and fixed to the other end of the rod. Relative to
the positive direction of the axis of rotation, the angular positions
of the sphere are negative, its angular velocity is negative, and its
angular acceleration is negative. The sphere is


1.   rotating clockwise and
     slowing down.                        71%
2.   rotating counterclockwise
     and slowing down.
3.   rotating clockwise and
     speeding up.
4.   rotating counterclockwise
     and speeding up.
5.   first rotating
     counterclockwise and then                  10%   10%         10%
     clockwise.
                                                             0%

                                           1     2      3     4         5
The angular speed of the minute
hand of a clock, in rad/s, is
                 35%
    1
       
1.    .
                             30%
  1800
    1                  25%

2. 60 .
      

3.    .
    1
      
   30
4. π.
5. 120π.
                                   5%   5%



                 1     2      3    4     5
The angular speed of the minute
hand of a clock, in rad/s, is
                 35%
    1
       
1.    .
                             30%
  1800
    1                  25%

2. 60 .
      

3.    .
    1
      
   30
4. π.
5. 120π.
                                   5%   5%



                 1     2      3    4     5
    Connections Between
Linear & Rotational Quantities
Angular Acceleration a
Comparison to 1-D Kinematics

      Angular                          Linear
     a  constant                   a  constant
     w  w0  a t                    v  v0  at
                   1                                 1 2
   q  q 0  w0 t  a t 2         x  x0  v0t        at
                   2                                 2
   w 2  w0 2  2a (q  q0 )        v2  v02  2a  x  x0 
   And for a point at a distance R from the rotation axis:

            x = Rq       v = wR          a = aR
By convention, q, w, a are positive if they are in the
counterclockwise direction.
               Decelerating Windmill
As the wind dies, a windmill that had
been rotating at w = 2.1 rad/s begins to
slow down at a constant angular
acceleration of a = 0.45 rad/s2.
How long does it take for the windmill
to come to a complete stop?




           w
    a av 
           t
          w   w f  wi (0)  (2.1 rad/s)
   t                                   4.7 s
        a av      a      (0.45 rad/s )
                                      2
The fan blade shown is slowing down.
Which option describes a and w?

                                   48%




                             24%         24%


 1.   w>0   and   a>0
 2.   w>0   and   a<0   5%
 3.   w<0   and   a>0
 4.   w<0   and   a<0   1     2     3     4
The fan blade shown is slowing down.
Which option describes a and w?

                                   48%




                             24%         24%


 1.   w>0   and   a>0
 2.   w>0   and   a<0   5%
 3.   w<0   and   a>0
 4.   w<0   and   a<0   1     2     3     4
         Rotational Kinematics




If the angular
acceleration is
constant:
The graphs below show angular velocity as a
function of time. In which one is the magnitude of
the angular acceleration constantly decreasing?




1.   a
                                           53%

2.   b                               42%



3.   c
4.   d                                           5%


5.   e
                           0%   0%

                           1    2     3     4     5
The graphs below show angular velocity as a
function of time. In which one is the magnitude of
the angular acceleration constantly decreasing?




1.   a
                                           53%

2.   b                               42%



3.   c
4.   d                                           5%


5.   e
                           0%   0%

                           1    2     3     4     5
                 Thrown for a Curve
To throw a curve ball, a pitcher
gives the ball an initial angular
speed of 157.0 rad/s. When the
catcher gloves the ball 0.795 s
later, its angular speed has
decreased (due to air resistance)
to 154.7 rad/s.
(a) What is the ball’s angular acceleration, assuming it to be constant?
(b) How many revolutions does the ball make before being caught?


        w  w0  a t
            w  w0   (157.0 rad/s)  (154.7 rad/s)
       a                                          3.03 rad/s 2
              t               (0.795 s)
        q  w0t  1 a t 2
                   2

             (157.0 rad/s)(0.795 s)  1 ( 3.03 rad/s 2 )(0.795 s) 2
                                       2

             123.9 rad  19.7 rev
                   Wheel of Misfortune
On a certain game show, contestants spin the
wheel when it is their turn. One contestant gives
the wheel an initial angular speed of 3.40 rad/s.
It then rotates through 1.25 revolutions and
comes to rest on BANKRUPT.
(a) Find the wheel’s angular acceleration,
assuming it to be constant.
(b) How long does it take for the wheel to
come to rest?


   w 2  w02  2a q
        w 2  w02      0  (3.40 rad/s) 2
   a                                       0.736 rad/s 2
          2q       2(2 rad/rev)(1.25 rev)
   w  w0  a t
        w  w0 0  (3.40 rad/s)
   t                           4.62 s
          a     (0.736 rad/s )
                             2
                     A Rotating Crankshaft
 A car’s tachometer indicates the angular velocity w of the crank shaft in rpm. A
car stopped at a traffic light has its engine idling at 500 rpm. When the light turns
green, the crankshaft’s angular velocity speeds up at a constant rate to 2500 rpm
in a time interval of 3.0 s.

How many revolutions does the crankshaft make in this time interval?




wi  500 rpm  (2 rad/rev)/(60 s/min)=52.4 rad/s w f  2500 rpm  5wi  262.0 rad/s

     w f  wi       (262.0 rad/s  52.4 rad/s)
a                                             69.9 rad/s 2
        t                   (3.0 s)
q f  qi  wi t  1 a  t 
                                2
                   2

                                                                          1rev
    0  (52.4 rad/s)(3.0 s)  1 (69.9 rad/s 2 )  3.0 s   472 rad            75 rev
                                                         2
                               2
                                                                         2 rad
                           Time to Rest
A pulley rotating in the counterclockwise direction is
attached to a mass suspended from a string. The mass
causes the pulley’s angular velocity to decrease with a
constant angular acceleration a = 2.10 rad/s2.
(a) If the pulley’s initial angular velocity is w0 = 5.40
rad/s, how long does it take for the pulley to come to
rest?
(b) Through what angle does the pulley turn during this
    time?
(c) If the radius of the pulley is 5.0 cm, through what distance is the mass lifted?

  w  w0  a t

  t  (w  w0 ) / a  0  (5.40 rad/s) / (2.10 rad/s2 )  2.57 s

   q  w0t  1 a t 2
              2

        (5.40 rad/s)(2.57 s)  1 ( 2.10 rad/s 2 )(2.57 s) 2  6.94 rad
                                2


   s  q r  6.94rad  5.0 cm  34.7cm
                                CD Speed
CDs and DVDs turn with a variable w
that keeps the tangential speed vt
constant.
Find the angular speed w and the
frequency that a CD must have in order
to give it a linear speed vt = 1.25 m/s
when the laser beam shines on the disk
(a) at 2.50 cm from its center, and
(b) at 6.00 cm from its center.

       vt
    w
       r
                    (1.25 m/s) 50.0rad 1rev
  r  2.50 cm: w                             7.96 rps
                   (0.0250 m)     s     2 rad
                    (1.25 m/s) 20.8 rad 1rev
  r  6.00 cm: w                              3.31 rps
                   (0.0600 m)     s      2 rad
   Rotational vs. Linear Kinematics
Analogies between linear and rotational kinematics:
    Connections Between
Linear & Rotational Quantities
     More Connections Between
    Linear & Rotational Quantities
This merry-go-round has
both tangential and
centripetal acceleration.

 at  ra        acp  rw 2  vt2 / r

 a  at2  acp
            2
                                       Speeding up


   tan   1
                acp   / at 
                    The Microhematocrit
 Suppose the centrifuge is just starting up, and
 that it has an angular speed of 8.00 rad/s and
 an angular acceleration of 95.0 rad/s2.
 (a) What is the magnitude of the centripetal,
 tangential, and total acceleration of the
 bottom of a tube?
 (b) What angle does the total acceleration
 make with the direction of motion?
                               2
     v2               rad 
 ac   w 2 r   8.00        9.07cm   580.5 s 2
                                                 cm
     r                 s 

             rad 
 aT  a r   95 2   9.07cm   861.7 cm 2
                s                       s

a  ac  aT 
       2        2
                     580.5 cm s   861.7 cm s 
                                   2 2            2 2
                                                         1039 cm s 2
         a       580.5 cm 
   tan  c
       1
                 
                 
                          s2
                                34  0.593rad
                                    o

          aT     861.7 s 2 
                          cm
                              

				
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