1) What is the magnitude of the angular displacement of the minute hand of a clock in 20 minutes?

2) Obtain the relation between linear velocity and angular velocity in uniform circular motion.

3) A body of mass 1 kg is tied to a string and revolved in a horizontal circle of radius 1 m. What is the maximum number of r.p.m. made so that the string does not break?
[Breaking tension of the string is 9.86 N]

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2014-09-25T19:13:31+05:30

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1)
    The Angular velocity of the minutes hand =  angle covered / time duration taken
         ω  =  360° / 1 hour = 360° / 3600 seconds = 0.1 ° / second 
            or   = 2 π radians / 3600 sec  = 0.00174 radians/sec
        
      Anglular displacement =  Ф = ω t =  (2 π / 3600 ) * 20 * 60 = 2π/3  radians 
                                           = 2*180/3 = 120 degrees

      In one hour the hand covers a full circle and so In 20 minutes the minutes hand covers 1/3 of full circle = 2π/3 radians.

2) 
      In uniform circular motion, the angular velocity ω is constant.  the linear velocity v is tangential to the circular arc at any time.  Linear speed has the same magnitude. The distance traveled by the particle is equal to the length of the arc.  let radius of circular arc be r.

     speed v = distance/ time  =  2 π r / T        so, T = 2π r / v
      ω = angle covered in a full circle / time to travel one full circle 
           = 2 π /  T  = 2 π / [ 2 π r / v ]    =  v / r
       v = r ω 
      Linear velocity = radius * angular velocity

    ANOTHER WAY:
   
           Let linear speed be v. Angle covered in time t be Ф.  
            Then distance traveled = arc length =  r Ф
                  so linear speed = v =   distance / time =  r Ф / t  = r  ω
                        as ω = Ф / t  = angle covered / time taken
            v = r ω  

3)
     The tension T in the string is = centripetal force  = m v²/r = m r ω²
         n = revolutions per minute  
           ω = 2π n / 60
 
         T =  m r (2π n / 60)²
 
         9.86 N = 1 kg * 1 meter * ( 2 π n / 60)²
    
         n² = 9.86 * 60² / 4 π² 

          n = 900

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