Final answer:
The hanging mass will move over a vertical distance of approximately 12.857 meters in 3.0 seconds.
Step-by-step explanation:
First, we need to calculate the moment of inertia of the pulley wheel. The moment of inertia of a solid disk is given by the formula I = (1/2) * m * r^2, where m is the mass of the disk and r is the radius.
Plugging in the values, we get I = (1/2) * 4.22 kg * (1.30 m)^2
= 3.4336 kg·m^2.
Next, we can calculate the acceleration of the system using Newton's second law, F = m * a.
The force that causes the acceleration is the tension in the string. The tension in the string is equal to the weight of the hanging mass, which is given by T = m * g, where m is the mass of the hanging mass and g is the acceleration due to gravity.
Plugging in the values, we get T = 1.74 kg * 9.8 m/s^2
= 17.052 kg·m/s^2.
Now we can calculate the acceleration of the system: 17.052 kg·m/s^2 = (4.22 kg + 1.74 kg) * a.
Solving for a, we get a = 17.052 kg·m/s^2 / 5.96 kg ≈ 2.8604 m/s^2.
Finally, we can calculate the distance the hanging mass moves in 3.0 seconds using the equation of motion, d = v0 * t + (1/2) * a * t^2, where v0 is the initial velocity of the hanging mass (which is 0 m/s because it starts from rest), a is the acceleration of the system, and t is the time.
Plugging in the values, we get d = 0 + (1/2) * 2.8604 m/s^2 * (3.0 s)^2
= 12.857 m.