A rock of mass m is tied to a string of length L and swung in a horizontal circle of radius r. The string can withstand a maximum tension T_max before it breaks. 

(a) What is the maximum speed v_max the rock can have without the string breaking? 

(b) The speed of the rock is now increased to 3v_max. The original single string is then replaced by N pieces that are all identical to the original string. What is the minimum value of N required so that the strings do not break? Ignore the force of gravity on the rock.