To solve this problem, we need to analyze the forces acting on the block and the spring in equilibrium.

Step 1: Forces Acting on the Block

The block is pushed against the wall by the spring, and the goal is to prevent the block from falling. Here's a breakdown of the forces involved:

• Weight (Gravitational Force): $F_g = m \cdot g$

  • Where $m$ is the mass of the block (0.30 kg), and $g$ is the acceleration due to gravity (9.8 m/s²).

• Normal Force (N): The spring applies a force that acts perpendicular to the wall. This force is the normal force $N$, which is equal to the restoring force of the spring: $N = F_{\text{spring}} = k \cdot x$

  • Where $k = 110 \, \text{N/m}$ is the spring constant, and $x$ is the compression of the spring.

• Frictional Force: The static friction prevents the block from sliding down. The maximum frictional force is: $F_{\text{friction}} = \mu_s \cdot N$

  • Where $\mu_s = 0.42$ is the coefficient of static friction.

Step 2: Equilibrium Condition

To keep the block from falling, the frictional force must be at least equal to the gravitational force acting downward: $F_{\text{friction}} \geq F_g$ This gives us the condition: $\mu_s \cdot N \geq m \cdot g$

Substituting $N = k \cdot x$, we have: $\mu_s \cdot k \cdot x \geq m \cdot g$

Step 3: Solve for $x$ (Minimum Compression)

Now, solving for $x$: $x \geq \frac{m \cdot g}{\mu_s \cdot k}$

Substitute the known values: $x \geq \frac{(0.30 \, \text{kg}) \cdot (9.8 \, \text{m/s}^2)}{(0.42) \cdot (110 \, \text{N/m})}$

Step 4: Calculation

Let me compute that:The minimum compression of the spring needed to keep the block from falling is approximately $0.064 \, \text{m}$ or 6.4 cm.

Would you like further details or have any other questions?

Here are 5 related questions you might find interesting:

1. What happens if the coefficient of static friction is increased or decreased?
2. How would the spring compression change if the mass of the block increased?
3. How would this scenario change if the spring constant were different?
4. What is the relationship between the angle of the surface and the required compression?
5. Can the same analysis apply if the block is on an inclined plane?

Tip: Static friction always adjusts to resist motion up to its maximum value, determined by the normal force and the coefficient of static friction.