To find the work $W_{F_2}$ done by the applied force $\vec{F_2}$ on the box as it moves a distance $L$ along the incline, let's go over the forces and confirm the direction and calculation carefully.

Given Variables:

• $F_2$: Applied force along the incline.
• $m$: Mass of the box.
• $g$: Acceleration due to gravity.
• $\theta$: Angle of the incline.
• $L$: Distance the box moves along the incline.
• $\mu$: Coefficient of kinetic friction.

Step-by-Step Solution

1. Identify the Forces Acting on the Box:

   • Applied Force, $F_2$: This is given, and it acts along the incline.
   • Gravitational Force, $mg \sin \theta$: The component of gravity acting down the incline.
   • Frictional Force, $f_k$: This force opposes the motion of the box up the incline and is given by: $f_k = \mu N = \mu mg \cos \theta$ where $N = mg \cos \theta$ is the normal force.

2. Work Done by $F_2$:

   • The work done by $F_2$, $W_{F_2}$, is given by the product of the force $F_2$ and the displacement $L$ in the direction of $F_2$: $W_{F_2} = F_2 \cdot L$

3. Setting Up the Net Force:

   • If we’re only calculating the work done by $F_2$, we don’t need to add or subtract the effects of other forces directly here. However, to understand the context:
     • The total force that would oppose $F_2$ includes both the gravitational component down the incline, $m g \sin \theta$, and the frictional force, $\mu m g \cos \theta$.

4. Final Answer: The correct expression for the work done by the applied force $F_2$ on the box as it moves a distance $L$ along the incline is: $W_{F_2} = F_2 \cdot L$

This answer, $W_{F_2} = F_2 L$, assumes $F_2$ is applied along the incline and does not include any effects from gravitational or frictional forces in the direct calculation of the work done by $F_2$ alone.

Let me know if you need further clarification or have any additional questions!

Relative Questions

1. How would the work change if the applied force $F_2$ had a different angle from the incline?
2. What effect does the coefficient of friction $\mu$ have on the net work done by all forces on the box?
3. How would you calculate the total work done on the box by all forces along the incline?
4. What would happen to $W_{F_2}$ if the box were moved twice the distance $L$?
5. How would the answer change if there were no frictional force?

Tip

When calculating work done by a specific force, remember to only consider that force and its direction relative to the displacement.