Final answer:
The problem is a physics question about friction and mechanics, where the maximum mass ratio m/m for which the block does not slide is equal to the coefficient of friction µ. To prevent the block from sliding, this mass ratio must not be exceeded.
Step-by-step explanation:
The physics problem described involves mechanics and friction. It pertains to a scenario where a small object slides inside a smooth hemispherical groove of a block on a rough surface. We want to find the maximum ratio of the mass of the small object to the block (m/m) such that the block does not slide on the horizontal surface.
When the small object is at the top, the net force in the horizontal direction that could cause the block to slide is zero. As the object slides down, it will exert a horizontal force on the block due to its circular motion. This force reaches its maximum when the object is at the bottom of the groove, which corresponds to the normal force due to the object's weight (m*g). The block will start sliding if this force exceeds the maximum static friction force the ground can apply on the block, which is µ * m * g, where µ is the coefficient of static friction between the block and the surface.
Setting the forces equal and solving for m/m gives us the critical ratio, m/m = µ. Since there's no mention of a coefficient of static friction, we assume µ refers to the coefficient of kinetic friction. Therefore, for the block not to slide, the mass ratio must not exceed the coefficient of friction µ.