In the figure above, a mass of 65.3 kg is attached to a light string, which is wrapped around a cylindrical spool of radius 0.47 m and moment of inertia 5.24 kg m2. The spool is suspended from the ceiling, and the mass is then released from rest a distance 6.32 m above the floor. How long does it take to reach the floor?Express your answer in seconds, to at least two digits after the decimal point.

To solve this problem, we can use the principle of conservation of mechanical energy.

Step 1: Calculate the potential energy of the mass at the initial position. The potential energy (PE) is given by the formula PE = mgh, where m is the mass (65.3 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (6.32 m). PE = (65.3 kg)(9.8 m/s^2)(6.32 m) = 4,058.33 J

Step 2: Calculate the final kinetic energy of the mass. The final kinetic energy (KE) is given by the formula KE = (1/2)mv^2, where m is the mass (65.3 kg) and v is the final velocity. Since the mass starts from rest, the initial velocity is 0 m/s. KE = (1/2)(65.3 kg)(v^2)

Step 3: Calculate the change in potential energy. The change in potential energy is equal to the work done by the gravitational force, which is given by the formula ΔPE = mgh. In this case, the change in potential energy is equal to the initial potential energy since the final height is 0. ΔPE = 4,058.33 J

Step 4: Calculate the change in kinetic energy. The change in kinetic energy is equal to the work done by the net force acting on the mass. In this case, the only force acting on the mass is the tension in the string, which is always perpendicular to the displacement. Therefore, the work done by the tension is zero, and the change in kinetic energy is also zero. ΔKE = 0

Step 5: Apply the principle of conservation of mechanical energy. According to the principle of conservation of mechanical energy, the total mechanical energy (PE + KE) remains constant throughout the motion. Therefore, the initial mechanical energy (PE) is equal to the final mechanical energy (KE). PE = KE 4,058.33 J = (1/2)(65.3 kg)(v^2)

Step 6: Solve for the final velocity. Rearranging the equation, we get: v^2 = (2)(4,058.33 J) / (65.3 kg) v^2 = 124.16 m^2/s^2 v = √(124.16 m^2/s^2) v = 11.14 m/s

Step 7: Calculate the time taken to reach the floor. The time taken to reach the floor can be calculated using the equation v = d/t, where v is the final velocity (11.14 m/s) and d is the distance traveled (6.32 m). t = d / v t = 6.32 m / 11.14 m/s t = 0.567 seconds

Therefore, it takes approximately 0.57 seconds for the mass to reach the floor.