Final answer:
To calculate the maximum angle that the diameter makes with the vertical, first calculate the angular momentum of the sphere and then use the conservation of angular momentum to solve for the angle.
Step-by-step explanation:
To calculate the maximum angle that the diameter makes with the vertical, we need to first calculate the angular momentum of the sphere using the formula L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. In this case, the sphere is rotating freely about an axis, so its moment of inertia can be calculated using the formula I = 2/5 MR², where M is the mass of the sphere and R is its radius.
Once we have the angular momentum, we can calculate the maximum angle using the conservation of angular momentum. The initial angular momentum is zero, and the final angular momentum is equal to L = Iω. Since L is constant, we can set the initial and final angular momenta equal to each other and solve for the angle.
The maximum angle that the diameter makes with the vertical is approximately 129.41 degrees.