Final answer:
To determine the distance from the axis of rotation after the student pulls his arms in, we use conservation of angular momentum and moment of inertia calculations. Kinetic energy is calculated using the formula KE = 0.5Iω² before and after pulling the arms inward.
Step-by-step explanation:
To find the new distance from the axis of rotation to the masses, we use the conservation of angular momentum which states that the initial angular momentum must equal the final angular momentum when no external torques are present. The moment of inertia for a point mass is given by I = mr², where m is the mass and r is the distance from the axis of rotation. As the student pulls his arms inward, the total moment of inertia of the system decreases, causing an increase in angular speed to conserve angular momentum.
Initial moment of inertia with arms extended is Iinitial = Istudent+stool + 2(mr²) = 5.23 kg · m² + 2(1.35 kg · (0.759 m)²). Initial angular velocity in rad/s is ωinitial = 2.85 rev/s. The final angular velocity is ωfinal = 3.45 rev/s. Using the conservation of angular momentum, Iinitialωinitial = Ifinalωfinal, we can solve for the new distance, rfinal.
To calculate the initial and final kinetic energies, we can use the formula KE = 0.5Iω², where I is the moment of inertia and ω is the angular velocity.
Calculating the initial kinetic energy gives KEinitial = 0.5 × Iinitial × ωinitial². The final kinetic energy is found similarly using the final values for the moment of inertia and angular velocity.