Final answer:
The length of the rope wound around the solid cylinder is calculated based on the ratio of the moments of inertia and equal angular accelerations for both cylinders, resulting in a length of approximately 221.24 m.
Step-by-step explanation:
To determine the length of the rope wound around the solid cylinder, we must use the concept of angular acceleration (α) and the moment of inertia (I). The torque (τ) caused by the force on each cylinder can be calculated using the formula τ = r * F, where r is the radius and F is the force applied. For the hollow cylinder, the moment of inertia is Ihollow = mhollow * r2, whereas for the solid cylinder, Isolid = msolid * r2 / 2, because for a solid cylinder, I = (1/2) * m * r2. Given that the solid cylinder is 20 times heavier than the hollow one, and that they acquire equal angular accelerations, the torque and therefore the product of force and radius must also be equal. Therefore, rsolid * F = 20 * rhollow * F, as F is the same for both. Since the radius is the same, the lengths of the ropes must be in the ratio of the number of turns, which is 4 for the hollow and we need to find x for the solid; hence, 4 / x = 1 / 20, leading to x = 80. So, the length of rope on the solid cylinder must be 80 turns, which is 80 times the circumference of the cylinder (2 * π * r). So, the length is 80 * 2 * π * 0.14 m, which is 70.4π m or approximately 221.24 m.