Solving Problems Involving Angular Velocity

 

Among the challenging problems I encounter  in Trigonometry are problems involving angular velocity. In this hub, I presented several problems involving angular velocity with their solution.

 

 

Problem Number One :

 

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.

 

Given :

 

Radius ( r )   =  1.5/2   =   .75 m =è   .75 (1/1000)   =   .00075 km

 

Linear Velocity (V)  =    24 kph

 

Angular Velocity (W)  =   ?   in revolution  per minute

 

V  =   r W =è      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

 

 

Problem Number Two :

 

A man was jogging on an oval track with radius  89 meters. If the man was jogging at a speed of 25 m/min, what was the central angle generated by the man after 5   minutes ?

 

Given :

 

Radius ( r ) =   89 m

 

Linear Velocity ( V )  = 25 m/min

 

Time  ( t )  =   5  minutes

 

W  =   ө / t        W  =   V/ r

 

W =  ( 25m/min) ( 1/89m)  =   25/89   rad/ min   =  0.281  rad/min

 

ө  = Wt

 

ө  =  (.281 rad/min)  (5 min)  -  1.4 radian

 

 Convert 1.4 radian to degrees :  ( 1.4 )  ( 180/∏ ) =  80.21 degrees

 

 

Problem Number Three :

 

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 ?

 

Given :

 

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

 

 

Problem  Number  Four :

 

Find the radius of a pulley which is driven at  10 rev/sec by a belt moving at 10m/sec.

 

Given :

 

W =    10  rev/sec

 

V  =   10  m/sec

 

Radius  ( r )  =   ?

 

 R = V/W

 

W  =  (10 rev/sec) (2 ∏ rad/rev ) = 20 ∏ rad/sec

 

     r  =   (10m/sec ) /(20 ∏rad/sec)  =  0.16 m

 

 

Problem Number Five :

 

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

 

 

Problem Number  Six :

 

Find the linear velocity of the tip of the minute hand of a clock if the hand is 7cm

Long ?

 

V  =  S / t

 

Given :

 

Radius  ( r )  =   7cm

 

W  =  1 rev/hr   =  2∏ rad/hr

 

V  =  ?

 

V  =  r W

 

V  =  (7 cm)  (2∏rad/hr)  =  14 ∏ cm/hr  = ( 14∏/60) cm/min  = .73 cm/min