Balls in a semicircle!

Let $\mu ≡ \frac{M}{N}$ be the mass of each ball in the semicircle.

We need the deflection angle in each collision to be $\theta = \frac{π}{N}$ .

However, if the ratio $\frac{\mu}{m}$ is too small, then this angle of deflection is not possible. We know that the maximum angle of deflection in each collision is given by $\sin \theta = \frac{\mu}{m}$ . Since we want $\theta = \frac{\pi}{N}$ here, this $\sin \theta ≤ \frac{\mu}{m}$ condition becomes (using $\sin \theta ≈ \theta$ , for the small angle $\theta$ )

$\theta ≤\frac{\mu}{m}$

$⇒\frac{\pi}{N} ≤ \frac{M/N}{m}$

$⇒ \pi ≤ \frac{M}{m}$ .