The problem involves calculating the force $P$ needed to cause impeding motion of block $B$ to the left and right under static friction conditions. Here's a breakdown of the provided information and how to approach the solution:

Given:

• Weight of Block $A = 200 \, \text{N}$
• Weight of Block $B = 100 \, \text{N}$
• Coefficient of static friction, $\mu_s = 0.20$

Objective:

1. Compute the force $P$ to cause impending motion of block $B$ to the left.
2. Compute the force $P$ to cause impending motion of block $B$ to the right.
3. Determine if a force $P = 50 \, \text{N}$ applied to the left will cause motion, and if so, which block moves.
4. Determine if a force $P = 150 \, \text{N}$ applied to the left will cause motion, and if so, which block moves.

Solution Steps:

1. Free Body Diagram Analysis:

   • Consider the forces acting on each block separately.

   • For Block $A$ on the inclined surface:

     • Normal force: $N_A = W_A = 200 \, \text{N}$
     • Friction force: $f_A = \mu_s \times N_A = 0.20 \times 200 = 40 \, \text{N}$

   • For Block $B$:

     • Normal force: $N_B = W_B = 100 \, \text{N}$
     • Friction force: $f_B = \mu_s \times N_B = 0.20 \times 100 = 20 \, \text{N}$

2. Calculating Force $P$ for Block $B$ Moving Left:

   • The force $P$ needs to overcome the static friction force on Block $A$ and also initiate movement for Block $B$.
   • Total friction to overcome: $F_{\text{friction, total}} = f_A + f_B = 40 + 20 = 60 \, \text{N}$
   • Therefore, $P = 60 \, \text{N}$ is needed to cause block $B$ to move left.

3. Calculating Force $P$ for Block $B$ Moving Right:

   • Similarly, for block $B$ to move right, the force $P$ must again overcome the total friction force: $P = 60 \, \text{N}$

4. Analysis for $P = 50 \, \text{N}$:

   • Since $P = 50 \, \text{N} < 60 \, \text{N}$, the applied force is not sufficient to overcome the total friction. No motion will occur.

5. Analysis for $P = 150 \, \text{N}$:

   • Since $P = 150 \, \text{N} > 60 \, \text{N}$, the applied force is more than enough to overcome the total friction. Block $B$ will move to the left.

If you would like a more detailed breakdown of any of these steps or have specific questions, let me know!

Further Related Questions:

1. What would be the force required if the coefficient of static friction were different?
2. How would the result change if Block $A$ were on a different inclined angle?
3. What happens if there is kinetic friction involved after the blocks start moving?
4. How to calculate the acceleration of the blocks once they start moving?
5. How does the force of $P$ change if Block $B$ is heavier than Block $A$?

Tip:

Always remember to break the problem into smaller steps and analyze each block or component separately using free-body diagrams. This helps in clearly visualizing all forces involved.