The problem given is:

"A vertical force $P$ at $A$ and another vertical force $F$ at $B$ produce a resultant of $100 \, \text{lb}$ down at $D$ and a counter-clockwise couple $D$ at $200 \, \text{lb} \cdot \text{ft}$. Find the magnitude and direction of force $P$ and $F$."

Solution Outline:

1. Identify Given Values:

   • Resultant force: $R = 100 \, \text{lb}$ (downward).
   • Couple moment: $M = 200 \, \text{lb} \cdot \text{ft}$ (counter-clockwise).

2. Analyze Equilibrium Equations:

   • Since $R$ acts downward, the forces $P$ and $F$ should sum up to give this resultant in the vertical direction.
   • The couple moment generated by $P$ and $F$ must match the given moment of $200 \, \text{lb} \cdot \text{ft}$.

3. Calculate Force Magnitudes:

   • Set up equilibrium equations to solve for $P$ and $F$ in terms of their distances from a reference point.
   • Use the provided couple and resultant to relate these forces and solve for their exact values.

Would you like a step-by-step calculation or more details on specific aspects?

Follow-Up Questions:

1. What is the principle of moments, and how does it apply here?
2. How would the problem change if the couple moment were clockwise?
3. Could we solve this with vector components if $P$ and $F$ were not vertical?
4. What additional information would we need if point locations $A$ and $B$ were unspecified?
5. How would adding another force at point $C$ impact the calculations?

Tip:

When dealing with forces and moments, always consider both equilibrium of forces (sum of forces) and equilibrium of moments (sum of moments).