# Solution Outline ## Given Data - \( F = F_1 + F_2 \) - \( F_{1b} + F_{2b} = 0 \) - \( F_1 = 80\,N \) - \( \theta = 30^\circ \) A triangle is drawn with: - Sides: \( a, b, c \) - Angles: \( \alpha, \beta, \phi \) - Law of sines: \[ \frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \phi}{c} \] ## Interpretation You are to **find the value of \( F_2 \) and the resultant force \( F \)**, given \( F_1 \) and the angle between them (\( \theta \)). ## Step-by-Step Solution ### 1. **Triangle Law of Forces** Let \( F_1 \) and \( F_2 \) be forces at an angle \( \theta \). - The resultant \( F \) can be found using the law of cosines: \[ F^2 = F_1^2 + F_2^2 + 2F_1F_2 \cos(\theta) \] - The triangle law (law of sines) applies: \[ \frac{F}{\sin \theta} = \frac{F_2}{\sin \alpha} = \frac{F_1}{\sin \beta} \] ### 2. **Component Balance (Equilibrium)** If \( F_{1b} + F_{2b} = 0 \), the forces are balanced in the \( b \) direction (perpendicular or a specific axis). ### 3. **If \( F_2 \) is Unknown** If you need \( F_2 \), you need either another angle or force value. The problem as shown does **not provide enough information** to solve for \( F_2 \) directly unless you assume certain standard triangle configurations (e.g., equilibrium, where all forces form a closed triangle). --- ## **Summary of Approach** 1. **Draw the force triangle:** - Sides: \( F_1, F_2, F \) - Angle between \( F_1 \) and \( F_2 \): \( \theta = 30^\circ \) 2. **Use Law of Sines or Law of Cosines:** - If another angle or side is known, use law of sines: \[ \frac{F}{\sin \theta} = \frac{F_2}{\sin \alpha} = \frac{F_1}{\sin \beta} \] - If only \( F_1, \theta \) are given, more information is needed. --- ## **What is Missing?** - The value or direction of \( F_2 \) or the resultant \( F \), or another angle. --- ## **Conclusion** **With the provided information, you cannot numerically solve for \( F_2 \) and \( F \) unless one more piece of data is given.** If you have additional information (like the direction of \( F_2 \) or the resultant \( F \)), please provide it. Otherwise, use the equations above to relate the values.