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.