https://file.aiquickdraw.com/mathbot/file/1726739351215y15k5v4q.pdf

The file contains a series of tutorial problems in mechanics. Here's a breakdown of the problems:

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### Problem 1
**Express forces \( P_1 \) and \( P_2 \) in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \).**  
- **Answer:** \( P_1 = -192\mathbf{i} - 422\mathbf{j} \, \text{N} \), \( P_2 = 1.28\mathbf{i} + 0.96\mathbf{j} \, \text{kN} \).

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### Problem 2
**Determine the resultant \( R \) of two forces acting on the hook using graphical, geometrical, and algebraic methods.**  
- **Answer:** \( R = -6.7\mathbf{i} + 10\mathbf{j} \, \text{kN} \). Alternatively, \( R = 12 \, \text{kN}; \theta = 124^\circ \).

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### Problem 3
**Resolve the force \( P \) into components along axes \( a \) and \( b \).**  
- **Answer:** \( P_a = 6 \, \text{N}; P_b = 16 \, \text{N} \).

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### Problem 4
**Determine the magnitude of \( P \), given \( P_x = 200 \, \text{N} \).**  
- **Answer:** \( P = 224 \, \text{N} \).

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### Problem 5
**Replace a 600 N force with a 400 N force in the negative \( y \)-direction and another force \( F \). Find the magnitude and direction of \( F \).**  
- **Answer:** \( F = 872 \, \text{N}; \theta = 53.4^\circ \).

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### Problem 6
**Replace force \( P \) with forces \( P_1 \) and \( P_2 \). \( P_1 \) is along the \( a-a \) direction, and \( P_2 = 42 \, \text{kN} \). Find \( P_1 \) and the direction of \( P_2 \).**  
- **Answer:** Two possible solutions:  
  - \( P_1 = 38 \, \text{N}; \theta = -52^\circ \)  
  - \( P_1 = 7 \, \text{N}; \theta = -8^\circ \).

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### Problem 7
**Express the resultant \( R \) of two forces acting on the frame in vector form. If \( R \) is applied through point \( E \), determine the vertical distance \( s \) of \( E \) from the base.**  
- **Answer:** \( R = 13\mathbf{i} + 2.5\mathbf{j} \, \text{kN} \), vertical distance \( s = 2.5 \, \text{m} \).

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Do you need detailed step-by-step solutions for any of these problems?

### Five follow-up questions:
1. How do you break a force into its \( i \)- and \( j \)-components?
2. What is the geometric method for finding the resultant of two forces?
3. How do you resolve forces into non-orthogonal axes?
4. Why is the magnitude of the force \( P \) greater than the given component \( P_x \)?
5. How do you use vector addition to replace forces?

**Tip:** Always check if the forces are perpendicular when resolving or summing forces, as it simplifies calculations.

Explore detailed solutions to mechanics tutorial problems including force vector components, resultant forces, and force resolution along different axes. Suitable for university-level mechanical engineering students.

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