Minimum Length of Cart + Pure Rolling!

A uniform disc of mass \(m = 12 \text{ kg}\) slides down along smooth, friction less hill, which ends in a horizontal plane without break. The disc is released from rest at a height of \(h = 1.25 \text{ m}\) (it has no initial speed and it does not rotate), and lands on the top of a cart of mass \(M = 6 \text{ kg}\), which can move on a friction less surface. The coefficient of kinetic friction between the cart and the disc is \(\eta_k = 0.4\). Find minimum length of the cart (in \(\text{ m}\)) so that the disc begins to roll without slipping before loosing contact with the cart.

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