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DYNAMICS - EXAMPLE


Ball on Cord
  Example

 

A 1 kg steel ball is connected with a cord that wraps around a fixed cylinder centered at O. At point 1, the ball's velocity, v1, is 10 m/s. What is the velocity when the ball reaches point 2 if there is a constant tension in the cord of 6 N for 2 seconds? Assume no friction or air resistance, and that it takes more than 2 seconds to go from point 1 to point 2.

   
    Solution


Cord Wrapping around Cylinder

 

The change in angular momentum of the ball rotating around the center cylinder will be effected by the tension in the cord. That tension will provide an angular impulse. This effect is modeled by the Angular Momentum Equation as,

     

     
   

The negative sign is due to the moment acting in the clockwise direction (acts on the ball).

The cord lengths are
     L1 = 6 m
     L2 = L1 - πr = 6 - π (2/π) = 4 m

Substitute known values, and integrate gives,
     -T r 2 = 1(4) v2 - 1(6)(10)
     -6 ( 2/π ) 2 = 4 v2 - 60

v2 = 13.09 m/s

It needs to be noted that this problem is a simplification of the real situation. The time, 2 seconds, is only an estimate, and the real time will be a function of the velocity which is not known. Also, the tension is a complex function of the cord length and angular velocity, and will not be constant. This simple looking problem is actually complex and beyond this introduction to angular momentum.