Rotating Rod Acts As A Ball Launcher

An infinitely thin rod of length \[L=10\text{ m}\] has a pivot at the indicated place on the above diagrams. The rod currently is making a \[\theta= 90^{\circ}\] angle with the ground. A ball of radius \[r=1\text{ m}\] is located tangent to the ground and the rod, as indicated in diagram \(A\). The rod starts rotating clockwise at a constant speed of \[\omega= 1^{\circ}/\text{s}\] When the upper tip of the rod is tangent with the ball, as indicated in diagram \(C\), the velocity of the ball can be expressed as \[\dfrac{p}{q}\text{ m}/\text{s}\] for positive coprime integers \(p,q\). What is \(p+q\)? \[\quad\] \(\text{Details and Assumptions}\)

There is no friction between any of the surfaces.

If necessary, assume that the pivot moves along with the ball in order for the rod to always have contact with the ball.

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