\(N\) identical balls lie equally spaced in a semicircle on a frictionless horizontal table, as shown. The total mass of these balls is \(M\). Another ball of mass \(m\) approaches the semicircle from the left, with the proper initial conditions so that it bounces (elastically) off all \(N\) balls and finally leaves the semicircle, heading directly to the left.

In the limit \(N → \infty\) (so the mass of each ball in the semicircle, \(\frac{M}{N}\), goes to zero), find the minimum value of \(\frac{M}{m}\) that allows the incoming ball to come out heading directly to the left.

Hint - Use the result obtained in Maximum deflection angle!

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