Final answer:
To determine the final kinetic energy of the elongated rotating ball system, we must use the conservation of angular momentum and the formula for rotational kinetic energy, taking into account the change in the moment of inertia due to the elongation of the string.
Step-by-step explanation:
When a ball rotates with a constant angular velocity and the length of the string is elongated by a factor of 1.5 times, we can determine the final kinetic energy by considering the conservation of angular momentum. The initial moment of inertia (Iinitial) is m * r2 and the final moment of inertia (Ifinal) is m * (1.5r)2. Using angular momentum conservation Linitial = Lfinal or Iinitial*ωinitial = Ifinal*ωfinal, where ω is angular velocity. However, we're not directly given the final angular velocity.
Instead, we need to apply the principle that kinetic energy in rotational motion is given by (1/2)Iω2. As the angular momentum is conserved but the radius increases, the final angular velocity will decrease. Therefore, we need to find the ratio of the initial and final angular velocities from the moment of inertia change, which can then be used to calculate kinetic energy. The final angular velocity (ωfinal) can be calculated from the relationship (Iinitial*ωinitial)/(Ifinal). Finally, the final kinetic energy can be found with the (1/2)Ifinalωfinal2 formula.