Resultant Force A8E4Ea

1. **State the problem:** We need to find the magnitude and coordinate direction angles of the resultant force given two forces $F_1$ and $F_2$ with magnitudes and angles depending on variables $z_1, z_2, z_3, z_4$. 2. **Given:** - $F_1 = 75 + z_1$ N - $F_2 = 55 + z_2$ N - $\theta_1 = 30 + z_3$ degrees - $\theta_2 = 60 + z_4$ degrees - $d = 1 + 0.1 z_1$ units 3. **Approach:** The resultant force $\vec{R} = \vec{F_1} + \vec{F_2}$. We find components of each force in $x$, $y$, and $z$ directions, then sum components. 4. **Coordinate system and components:** - $F_1$ points upper-left with components: - $F_{1x} = -F_1 \cos(\theta_2)$ (negative x-direction) - $F_{1y} = F_1 \sin(\theta_2)$ (positive y-direction) - $F_{1z} = d$ (given as $d$ in z-direction) - $F_2$ points lower-right with components: - $F_{2x} = F_2 \cos(\theta_1)$ (positive x-direction) - $F_{2y} = -F_2 \sin(\theta_1)$ (negative y-direction) - $F_{2z} = 0$ 5. **Calculate resultant components:** $$ R_x = F_{1x} + F_{2x} = -F_1 \cos(\theta_2) + F_2 \cos(\theta_1) $$ $$ R_y = F_{1y} + F_{2y} = F_1 \sin(\theta_2) - F_2 \sin(\theta_1) $$ $$ R_z = F_{1z} + F_{2z} = d + 0 = d $$ 6. **Magnitude of resultant force:** $$ R = \sqrt{R_x^2 + R_y^2 + R_z^2} $$ 7. **Coordinate direction angles:** The angles $\alpha$, $\beta$, and $\gamma$ with the $x$, $y$, and $z$ axes respectively are: $$ \alpha = \cos^{-1}\left(\frac{R_x}{R}\right), \quad \beta = \cos^{-1}\left(\frac{R_y}{R}\right), \quad \gamma = \cos^{-1}\left(\frac{R_z}{R}\right) $$ 8. **Summary formula:** $$ \begin{cases} R_x = -(75+z_1) \cos(60+z_4) + (55+z_2) \cos(30+z_3) \\ R_y = (75+z_1) \sin(60+z_4) - (55+z_2) \sin(30+z_3) \\ R_z = 1 + 0.1 z_1 \\ R = \sqrt{R_x^2 + R_y^2 + R_z^2} \\ \alpha = \cos^{-1}\left(\frac{R_x}{R}\right), \beta = \cos^{-1}\left(\frac{R_y}{R}\right), \gamma = \cos^{-1}\left(\frac{R_z}{R}\right) \end{cases} $$ 9. **Interpretation:** Plug in values for $z_1, z_2, z_3, z_4$ to compute numeric results. This completes the solution for the magnitude and coordinate direction angles of the resultant force.