Resultant Force 4E249F

1. **State the problem:** We need to find the magnitude and coordinate direction angles of the resultant force vector $\mathbf{F}_R = \mathbf{F}_1 + \mathbf{F}_2$ where $F_1 = 75+z_1$ N, $F_2 = 55+z_2$ N, $\theta_1 = 30+z_3$ degrees, $\theta_2 = 60+z_4$ degrees, and $d = 1+0.1z_1$ units. 2. **Understand the vectors:** $\mathbf{F}_1$ points up-left and $\mathbf{F}_2$ points down-right from the origin with given angles $\theta_1$ and $\theta_2$ relative to reference lines. 3. **Express forces in components:** Assuming the coordinate system with axes $x$, $y$, and $z$ as described: - $\mathbf{F}_1$ components: - $F_{1x} = -F_1 \cos(\theta_1)$ (left direction) - $F_{1y} = 0$ (no y component given) - $F_{1z} = F_1 \sin(\theta_1)$ (upward) - $\mathbf{F}_2$ components: - $F_{2x} = F_2 \cos(\theta_2)$ (right direction) - $F_{2y} = 0$ (no y component given) - $F_{2z} = -F_2 \sin(\theta_2)$ (downward) 4. **Calculate resultant components:** $$ F_{Rx} = F_{1x} + F_{2x} = -F_1 \cos(\theta_1) + F_2 \cos(\theta_2) $$ $$ F_{Ry} = 0 + 0 = 0 $$ $$ F_{Rz} = F_1 \sin(\theta_1) - F_2 \sin(\theta_2) $$ 5. **Calculate magnitude of resultant force:** $$ F_R = \sqrt{F_{Rx}^2 + F_{Ry}^2 + F_{Rz}^2} = \sqrt{\left(-F_1 \cos(\theta_1) + F_2 \cos(\theta_2)\right)^2 + 0 + \left(F_1 \sin(\theta_1) - F_2 \sin(\theta_2)\right)^2} $$ 6. **Calculate coordinate direction angles:** The angles between $\mathbf{F}_R$ and the axes are given by: $$ \alpha = \cos^{-1}\left(\frac{F_{Rx}}{F_R}\right) \quad \text{(angle with x-axis)} $$ $$ \beta = \cos^{-1}\left(\frac{F_{Ry}}{F_R}\right) \quad \text{(angle with y-axis)} $$ $$ \gamma = \cos^{-1}\left(\frac{F_{Rz}}{F_R}\right) \quad \text{(angle with z-axis)} $$ 7. **Important notes:** - Angles should be converted to degrees if using radians. - Angles must be between $-180$ and $180$ degrees. - Use $\cancel{...}$ to show canceled terms if simplifying fractions (not applicable here). 8. **Summary:** - Calculate $F_1$, $F_2$, $\theta_1$, $\theta_2$, and $d$ from given expressions. - Compute components $F_{Rx}$, $F_{Ry}$, $F_{Rz}$. - Find magnitude $F_R$. - Find angles $\alpha$, $\beta$, $\gamma$. This completes the solution steps for the first question.