To find the total work $W_{F_1}$ done on the block by the applied force $\vec{F_1}$ as it moves a distance $L$ up the incline, we consider the force components acting on the block.

1. Work Done by the Applied Force: The work done by the applied force $\vec{F_1}$ is given by: $W_{F_1} = F_1 \cdot L \cdot \cos(\alpha)$ where $\alpha$ is the angle between the applied force and the direction of motion. If $\vec{F_1}$ is parallel to the incline, $\alpha = 0$ and thus $\cos(0) = 1$.

   Hence, $W_{F_1} = F_1 \cdot L$

2. Components of Forces:

   • The gravitational force acting down the incline is $mg \sin(\theta)$.
   • The frictional force acting opposite to the direction of motion is $F_{\text{friction}} = \mu mg \cos(\theta)$.

3. Total Work Against Gravity and Friction: The work done against gravity and friction while moving up the incline is: $W_{\text{gravity}} = mg \sin(\theta) \cdot L$ $W_{\text{friction}} = \mu mg \cos(\theta) \cdot L$

4. Total Work Done: Thus, the net work done on the block considering these forces is: $W_{\text{net}} = W_{F_1} - W_{\text{gravity}} - W_{\text{friction}}$

   Substituting values gives: $W_{F_1} = F_1 \cdot L - (mg \sin(\theta) + \mu mg \cos(\theta)) \cdot L$

5. Final Expression: $W_{F_1} = L \left( F_1 - mg \sin(\theta) - \mu mg \cos(\theta) \right)$

This expression gives the total work done on the block by the applied force as it moves a distance $L$ up the incline.

Would you like more details on any part of this? Here are some related questions you might consider:

1. How does the angle of the incline affect the work done?
2. What happens to the work done if the friction coefficient changes?
3. How do you calculate the acceleration of the block?
4. What is the relationship between work and energy in this scenario?
5. How would the work done change if the block moved down the incline instead?

Tip: Always ensure to resolve forces into components parallel and perpendicular to the motion for accurate work calculations.