Assume the total length of the plane is , the coefficient of friction is and the mass of the particle is .
The acceleration of the particle at right angles to the plane is zero. So, resolving in this direction, the normal contact force between the particle and the plane is given by
and, since the particle moves at right angles to this force, it does a zero amount of work.
During the second stage the friction force acting on the particle is . In this phase, the particle moves through a distance down the plane, and so the work done on the particle by this friction force is therefore .
The particle falls through a total height . The work done on the particle by its weight is therefore .
Since the particle starts and finishes at rest, the gain in its KE is zero, and therefore the total amount of work done on the particle is zero. So we have: