To solve this problem, let’s break down the forces and analyze the static equilibrium conditions. Here’s a step-by-step guide.

Problem Information:

1. Weight of crate, $W = 250$ lbs
2. Distance to center of gravity $G$ from base $A$: $2.5$ ft vertically.
3. Dimensions of crate: $4.5$ ft tall and $3.0$ ft wide.
4. Coefficient of static friction $\mu_s = 0.4$.
5. Goal: Find the maximum horizontal force $P$ that can be applied without causing movement.

Steps for Solution

1. Identify forces and moments:

   • The crate is in static equilibrium, so the sum of all forces and moments must equal zero.
   • There are three forces acting on the crate:
     • The applied force $P$ horizontally at the upper left corner.
     • The weight $W = 250$ lbs acting downwards at point $G$.
     • The normal force $N$ acting upwards at point $A$.
   • There’s also a frictional force $f$ at point $A$, which resists the horizontal motion and is given by $f = \mu_s N$.

2. Set up equilibrium equations:

   Sum of Forces in the Vertical Direction ( $\Sigma F_y = 0$ ):

   $N = W = 250 \text{ lbs}$

   Sum of Forces in the Horizontal Direction ( $\Sigma F_x = 0$ ):

   $f = P \Rightarrow \mu_s N = P$ $P = \mu_s \times W = 0.4 \times 250 = 100 \text{ lbs}$

3. Moment about Point $A$: To ensure the crate doesn’t tip over, take moments about point $A$.

   Moment Equation:

   • The clockwise moment caused by $P$ is: $M_P = P \times 4.5 \text{ ft}$
   • The counterclockwise moment due to the weight $W$ at the center of gravity $G$ is: $M_W = W \times 1.5 \text{ ft}$ Setting the moments equal for static equilibrium: $P \times 4.5 = 250 \times 1.5$ $P = \frac{250 \times 1.5}{4.5} = \frac{375}{4.5} = 83.33 \text{ lbs}$

4. Conclusion: The limiting factor for the applied force $P$ is the tipping moment, giving us: $P = 83.33 \text{ lbs}$

So, the maximum force $P$ that can be applied to the crate without causing movement is 83.33 lbs.

Would you like further explanation or have any questions?

Related Questions:

1. How does the coefficient of friction affect the maximum force $P$ in this scenario?
2. What would happen if the crate's weight increased? How would that change the maximum force $P$?
3. How would the result change if the force $P$ were applied lower on the crate?
4. What role does the center of gravity play in determining tipping risk in static equilibrium problems?
5. How does the width of the crate affect the tipping point calculation?

Tip:

When solving static equilibrium problems, always start by identifying all forces, establishing coordinate axes, and writing down equilibrium equations for both forces and moments.