The problem involves determining the components of the force $F$ in the directions $u$ and $v$, where $F = 430 \, \text{lbf}$ and the angle between $F$ and the $v$-axis is $25^\circ$. The image shows that the coordinates of the $u$ and $v$ directions form a triangle with the x-axis and y-axis.

Approach:

1. Given force $F = 430 \, \text{lbf}$.
2. Given angle $\theta = 25^\circ$ between the force and the $v$-axis.
3. Use trigonometry to resolve the components along the $u$ and $v$-axes.

$u$-component (parallel to the x-axis direction):

• This is the horizontal component, calculated as: $F_u = F \cdot \cos(\theta)$

$v$-component (parallel to the y-axis direction):

• This is the vertical component, calculated as: $F_v = F \cdot \sin(\theta)$

Calculating:

Let's calculate the components.The components of the force $F = 430 \, \text{lbf}$ are as follows:

• $u$-component: $389.7123 \, \text{lbf}$
• $v$-component: $181.7259 \, \text{lbf}$

These values closely match the given components in the image.

Would you like further details on how these calculations work or any other clarifications?

Here are five follow-up questions to explore:

1. How are the $u$ and $v$-axes determined geometrically in force diagrams?
2. What is the significance of resolving forces into components in physics?
3. Can we calculate the angle between $F$ and the $u$-axis as well?
4. How would we calculate the resultant force if given the components instead?
5. What role does the angle play in determining the force's effect in each direction?

Tip: When resolving forces, make sure to use the correct angle reference for each axis to avoid errors in your calculations.