A Cub Cadet CC4125 String Trimmet has a 17\" cutting and a bumb head that allows

Question

A Cub Cadet CC4125 String Trimmet has a 17" cutting and a bumb head that allows for string extension. Two circular patterns are shown below. Suppose the cutting head rotates with constant angular velocity for both cutting patterns. Also suppose the cutting radii before and after the bump are given by, respectively t_1 = 3.5 inches (88.9 mm) and I_2 = 8.5 inches (215.9 mm). Let v_i (i = 1, 2) denote the linear speed of the string tip located a distance I_i from the center of the cutting head. Determine the boost in string - tip speed associated with the line extension, determine the ratio v_2/v_1. Illustration of a Cub Cadet CC4125 String Trimmer and the cutting pattern at two lengths (right image). On the right, the line spool is represented by a gray circle radius r. The line spool has a diameter d = 2r of 3 inches (76.2). The radius of the first (blue) cutting circle is I_1 = 3.5 inches (88.9 mm); the radius of the second (green) cutting circle is I_2 = 8.5 inches (215.9 mm) v_2/v_1 = 0.70 v_2/v_1 = 1.00 v_2/v_1 = 1.41 v_2/v_1 = 1.70 v_2/v_1 = 2.40

Explanation / Answer

we know that the centripetal force acts towards the centre

F = mv^2/r
ma = mv^2/r ==> a = v^2/r

relation between the angular and lienar accleration is a = r * alpha

given radii of inner and outer circles are l1/l2
   l1 = 3.5 inch = 8.89 cm , l2 = 8.5 inch = 21.59 cm

   r*alpha= v^2/r  
   v^2 = r^2*alpha ==> v = sqrt(r^2*alpha)= r*sqr(alpha)

if both having same angular accelration , then v2/v1 = r2/r1 = l2/l1 = 8.5/3.5= 2.42

so answer is option e ) v2/v1 = 2.42

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