To solve this problem, we need to determine the maximum force $P$ that can be safely applied at point $D$ such that the reaction at point $C$ does not exceed 250 lb.

Step-by-Step Solution:

1. Understanding the Problem:

   • The lever $BCD$ is hinged at $C$ and connected to a control rod at $B$.
   • The lever can pivot about point $C$, and a force $P$ is applied at point $D$.
   • We need to determine the maximum value of $P$ such that the reaction force at $C$ is limited to 250 lb.

2. Free Body Diagram (FBD) and Forces:

   • At hinge $C$, there will be a reaction force with both horizontal ($C_x$) and vertical ($C_y$) components.
   • Force $P$ is applied at $D$ in a horizontal direction.
   • The control rod at $B$ exerts a force, but since we are interested in finding the maximum $P$, we focus on moments about point $C$.

3. Taking Moments about Point $C$:

   • The moment arm for the force $P$ at $D$ is 7.5 in (vertical distance from $C$ to $D$).
   • The moment arm for the force at $B$ is 3 in (vertical distance from $C$ to $B$).

   The sum of moments about point $C$ is given by: $M_C = P \times 7.5 - F_B \times 3 = 0$

   Where $F_B$ is the horizontal force in the control rod at $B$.

   Solving for $F_B$, we have: $F_B = \frac{P \times 7.5}{3} = 2.5P$

4. Calculating the Reaction at Point $C$:

   • The vertical reaction component $C_y = 0$ since there are no vertical forces.
   • The horizontal reaction component $C_x$ will balance both $F_B$ and $P$:

   $C_x = F_B + P = 2.5P + P = 3.5P$

   The resultant reaction force at $C$ is: $R_C = C_x = 3.5P$

   Since the maximum allowable value for $R_C$ is 250 lb: $3.5P = 250$

   Solving for $P$: $P = \frac{250}{3.5} \approx 71.43 \text{ lb}$

Conclusion:

The maximum force $P$ that can be safely applied at $D$ is 71.43 lb.

Would you like more details on this problem or have any further questions?

Related Questions to Explore:

1. How do you calculate moments in mechanics problems?
2. What is the significance of the reaction forces at a hinge in a static problem?
3. How do different forces affect the equilibrium of a system?
4. How can you use trigonometry to solve mechanics problems involving angles?
5. What are the conditions of equilibrium in two-dimensional force systems?

Tip:

When solving problems involving moments, always choose a point where the maximum number of unknown forces intersect to simplify the calculations.