Physics problem

[Luc@s]

The energy needed to compress the spring is indeed
but 10 cm = 0.1 m ;) (

You should proceed step by step. First, what is the total energy E when the mass is dropped ? Its not moving at this very moment so K = 0. But it has some gravitational energy. If we choose the reference so that the gravitational energy down the slope is 0, then .

When the mass gets down the slope, the total energy is still the same as before () because there was supposedly no friction. The only difference is that the energy was entirely transformed under its kinetic form (E = K), because the mass reached a position where its potential energy is 0.

The energy required to move along the path is obtained from the work of the friction force which is now taken into account. This force is constant during the motion. Thus, so that at the end of the path

(F can be calculated from the weight of the object and the friction coefficient)

Now, the kinetic energy has to be greater than the elastic energy required (which you calculated already).