Topic 2.1 Extended K – Angular speed and velocity  Consider two times in a particle's circular motion: x y θ1θ1 ω = and the instantaneous angular speed ω as  We define the average angular speed ω as θ2θ2 t1t1 t2t2 θtθt θ 2 - θ 1 t 2 - t 1 = Average Angular Speed ω = lim  t  0 Instantaneous Angular Speed  Contrast angular speed ω =  θ/  t with linear speed v =  x/  t.  Angular speed is measured in radians per second (rad/s).  The time for one revolution (or cycle) is called the period T, and is measured in seconds per cycle or just seconds. θtθt = dθ dt

Topic 2.1 Extended K – Angular speed and velocity  How are the linear speed v of the particle and the angular speed of the particle related? x y θ  Recall the relationship for arc length s: t1t1 t2t2 s r s = rθ, Definition of Arc Length θ in radians  Then we can find v by as follows: v = stst = r θtθt v = rω Relation between Linear Speed and Angular Speed FYI: Since we are speaking of circular motion, r is constant.

Topic 2.1 Extended K – Angular speed and velocity  Why do we use angular speed, when we have linear speed to fall back on?  Observe the following animation:  Note that each point on the disk covers a different distance during the same time interval.  This means that every point on the disk has a different linear speed v.  But each point makes one revolution in the same time as every other point.  This means that every point on the disk has the same angular speed .

Topic 2.1 Extended K – Angular speed and velocity  We define the period T to be the time it takes an object to rotate one complete revolution (cycle).  Note that each point on the disk has the same period.  The period T is measured in seconds (or seconds per cycle).  We define the frequency f to be the number of revolutions (cycles) per second. f = 1T1T Relation Between Frequency and Period  The units for f are s -1 or Hertz (Hz).

Topic 2.1 Extended K – Angular speed and velocity  Recall that v = rω. v = 2rT2rT Why? rω = 2rT2rT Why?  Then if the angular speed ω is constant, v = rω.  But  So that  Therefore ω =ω = 2T2T  Which we can rewrite as ω = 2  f Why? Frequency and Angular Speed

Topic 2.1 Extended K – Angular speed and velocity  Of course, the difference between speed and velocity is direction.  For rotation, the direction of  is given by the right hand rule illustrated below: axis of rotation   You might be wondering why we don’t just say “clockwise” (CW) and “counterclockwise” (CCW).  The reason is simple: CW and CCW depend on a reference frame. For example, if you are viewing the rotation from here the rotation appears to be CCW.  If you are viewing the rotation from here the rotation appears to be CW.

Topic 2.1 Extended K – Angular speed and velocity An “old time” record player rotates a record at 33.3 rpm. (a) What is the angular speed (in rad/s) of the record? “rpm” stands for “revolutions per minute” ω =ω = 33.3 rev min 2  rad rev 1 min 60 s = 3.49 rad/s (b) What is the frequency of the rotation? ω = 2  f  f =f = ω2ω2 =  = 0.56 Hz (c) What is the period of the rotation? T = 1f1f = = 1.8 s (d) What is the speed of a point located 4 cm from the center of the record? v = rω = (0.04)(3.49) =.140 m/s Question: What would be the linear speed of a point at the CENTER?Question: What would be the angular speed of a point at the CENTER?