Let's break down and solve the problem step by step.

Given:

• A horizontal force $\vec{F} = 13 \, \text{N}$ pushes a block weighing $3.4 \, \text{N}$ against a vertical wall.
• Coefficient of static friction $\mu_s = 0.59$.
• Coefficient of kinetic friction $\mu_k = 0.38$.
• The block is not moving initially.

(a) Will the block move?

1. Forces Acting on the Block:

   • The weight of the block, $W = 3.4 \, \text{N}$, acts downward.
   • The applied force, $F = 13 \, \text{N}$, is horizontal and pushes the block against the wall.
   • The frictional force, $f_s$, resists the block's motion.

2. Condition for Static Friction: The maximum static friction is: $f_s^{\text{max}} = \mu_s F_{\text{normal}}$ where $F_{\text{normal}} = 13 \, \text{N}$ because the horizontal force pushing the block against the wall is the normal force.

   So: $f_s^{\text{max}} = 0.59 \times 13 = 7.67 \, \text{N}$

3. Compare Forces:

   • The block's weight is $3.4 \, \text{N}$, which acts downward.
   • The maximum static friction force is $7.67 \, \text{N}$.

   Since $f_s^{\text{max}} > W$, the static friction can hold the block in place, meaning the block will not move.

(b) Force on the block from the wall (in unit-vector notation):

• The force exerted on the block by the wall has two components:

  1. Horizontal Component ($F_x \hat{i}$):

     • This is the normal force, which is equal to the applied force $\vec{F}$.
     • So, $F_x = -13 \, \text{N}$ (negative because the wall pushes back in the negative $x$-direction).

  2. Vertical Component ($F_y \hat{j}$):

     • This is the static friction force, which balances the weight of the block.
     • So, $F_y = 3.4 \, \text{N}$ (positive in the upward direction).

Thus, the force on the block from the wall in unit-vector notation is: $\vec{F}_{\text{wall}} = -13 \hat{i} + 3.4 \hat{j}$

Would you like more details or have any questions? Here are 5 related questions to think about:

1. What would happen if the coefficient of static friction were lower?
2. How would the block behave if the applied force increased to 20 N?
3. What would be the effect of reducing the weight of the block?
4. How is the force of kinetic friction used in such problems?
5. How can you calculate the acceleration of the block if it started moving?

Tip: Always compare the forces acting on the block (friction, weight, applied force) to determine motion.