(Solved): How to solve? A steampunk mechanism guides a ball along a semicircular slot by rotating a lever clo ...

How to solve?

A steampunk mechanism guides a ball along a semicircular slot by rotating a lever clockwise at \( \omega=4.8 \mathrm{rad} / \mathrm{s} \). The path of the ball is given by \( r=D \cos (\theta) \), where the diameter of the slot is \( D=8.7 \mathrm{~cm} \) For small values of the angle, \( \theta \), the weight of the ball is nearly aligned with the slot, so the centripetal acceleration is mostly due to contact with the the outer rim of the slot. For large angles, when the ball is near the top of the semicircle, the inner rim of the slot counteracts the weight of the ball to keep it moving along the circular path. Find the transition where neither the outer rim nor the inner rim exert any forces on the ball: the angle where the component of the ball's weight along the lever exactly matches the centripetal force needed to guide the ball along the slot. Neglect the size of the ball and the effects of friction. The ball transitions from outer rim to inner rim guiding at an angle of degrees (round answer to 2 significant figures)