Solution - Swinging Ball on Rough Inclined Plane

There are two conditions that have to be fulfilled to make one full swing. One is to maintain the string in tension, and another one is to have enough energy to complete one-cycle friction.

For the first condition, consider when the ball has made an angle φ like picture below.
From the picture, we can get tension of the string,

In order to make the string in tension at all time, the minimum value of T must equal to zero.
The condition when T reach its minimum value,

Because , so

Now, consider the forces that are align to ball's velocity.

Substitute for , and we get

Substitute dv'/dt from equation (3) to equation (2),

Insert φ to equation (1)

By using conservation of energy,

Now, we need to find the minimum speed required for second condition (i.e. the ball has to have enough energy to complete one-cycle friction.
By using conservation of energy,

As we see, the speed required for the second condition is larger than the speed required for the first condition. So, we may conclude that the minimum speed required for both conditions is

That's my solution.
If you have any question, correction, or comment, please leave a comment below.
Thank you.