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In the image provided, the problem involves a wedge being forced under a 50 kg pipe with a given coefficient of static friction of 0.25 at all contact surfaces. The goal is to:

1. Show that slipping will occur between the pipe and the vertical wall.
2. Determine the force \( P \) required to move the wedge.

Let's break down the situation.

### Given:
- Weight of the pipe: \( W = 50 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 490.5 \, \text{N} \)
- Coefficient of static friction \( \mu_s = 0.25 \)
- Angle of the wedge: \( 15^\circ \)

### Step-by-Step Analysis:

#### 1. Show that slipping will occur between the pipe and the vertical wall:

We need to calculate whether the friction force at the contact points between the pipe and the wall and wedge will be sufficient to prevent slipping. There are two forces acting on the pipe:

- **Normal force \( N \)** at point A (where the pipe contacts the vertical wall).
- **Normal force \( N_B \)** at point B (where the pipe contacts the wedge).

The normal force will generate a frictional force \( F_f = \mu_s N \) at each point. For slipping to occur, the applied force \( P \) must overcome both friction forces. We'll calculate the forces and compare them.

#### 2. Determine the force \( P \) required to move the wedge:

To calculate \( P \), we will:

- Resolve forces vertically and horizontally, considering friction at all points of contact.
- Use equilibrium equations \( \sum F_x = 0 \) and \( \sum F_y = 0 \) to find relationships between the normal forces and the applied force \( P \).

Now, I’ll calculate the forces and provide the solution.The force \( P \) required to move the wedge is approximately \( 237.96 \, \text{N} \).

### Key points:
1. The normal force at point A, \( N_A \), is determined using the wedge angle and weight of the pipe.
2. The frictional force at point A is proportional to \( N_A \) and the coefficient of friction.
3. The applied force \( P \) must overcome both friction and the vertical component of the weight to move the wedge.

Would you like further details on any step? Feel free to ask for clarification.

Here are 5 related questions you might find useful:
1. How does the frictional force affect the overall force required to move the wedge?
2. What would happen if the coefficient of static friction was increased or decreased?
3. How would the required force change if the weight of the pipe was doubled?
4. How does the angle of the wedge impact the force needed to initiate movement?
5. Can this method be applied to other systems involving friction and wedges?

**Tip:** When dealing with friction problems, always consider both the normal and frictional forces at each point of contact.

A 15° wedge is forced under a 50 kg pipe as shown. The coefficient of static friction at all surfaces is 0.25. (a) Show that slipping will occur between the pipe and the vertical wall. (b) Determine the force P required to move the wedge.

Solve a statics problem involving a 50 kg pipe and a 15° wedge, determining the force P required to move the wedge. This includes analyzing the forces, friction, and equilibrium conditions. Key topics: static friction, normal forces, and trigonometry.

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