Angular Velocity

Definition : Angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating.


SI Unit  :  radians per second, degrees per second, revolutions per second, revolutions per minute.


Direction  : perpendicular to the plane of rotations.


Symbol : omega (ω, rarely Ω).


Equations :       

Angular Velocity of A Particle

Average Velocity = Circumference of the Circle/Time

Average Speed/Velocity = 2πr/T where, T is the period of the system and r is the radius of the revolution.

                              ω=2π/T=2πf

 where, f is frequency and T is the period

Example  1 : 

A calesa with wheels whose diameter is 1.5 meters is traveling at 24 kph. Find the angular velocity of the wheel  in revolutions per minute.

Radius ( r )   =  1.5/2 

                      = 0 .75 (1/1000)   

                      =   0.00075 km

Linear Velocity (V)  =    24 kph

Angular Velocity (W)  =   ?   in revolution  per minute

 V  =   r W     

 W  =   V/r

W  =  (24 km/hr )(1/.00075 km)   =   32,000 radians  per hour

To convert to revolution per minute :

(32,000  rad/hr) (1 rev/2∏) (1 hr/60 min)  = 32,000/376.992   =   84.88

Example 2 : 

Suppose a point on a circle with radius  6  cm moves around a circle with angular velocity  of  2∏/5  rad/sec. What is the length of the arc generated after 10 seconds ?

 Radius  ( r )  =  6 cm

 Angular Velocity (W )  = 2∏/5  rad/sec

 Time ( t )   =  10  seconds

 S  =   ?  arc length

V  = r W

    =   (6 cm ) (2) (3.1416)/5  

    =   7.54  cm/sec

 S  =  Vt   

 S  =  ( 7.54 cm/sec) (10 sec) 

     =   75.4  cm

Example 3 : 

A ferriswheel has a diameter of  10 meters. It is rotating at the rate of  500 m/min. Find its angular velocity in rad/sec.

Radius ( r )  =  10/2 

                    = 5 m

 Velocity (V ) = 500 m/min

 W  =  ?

W  =   V/r

W  =   (500m/min )/5m 

     =  100 rad/min

     = (100 rad/min) (1 min/60 sec) 

     = 100/60 rad/sec

     =  1.7 rad/sec