Resultant Force 19D2F6

1. **Problem Statement:** 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. **Goal:** Find the magnitude $R$ and coordinate direction angles $\alpha, \beta, \gamma$ of the resultant force $\vec{R} = \vec{F_1} + \vec{F_2}$. 4. **Step 1: Express forces in components.** Assuming the coordinate system as shown, and using the angles $\theta_1$ and $\theta_2$ to find components: - For $F_1$, components depend on $d$ and 1 forming a right triangle, so the direction cosines can be found. - For $F_2$, direction angles are given. 5. **Step 2: Calculate components of $F_1$.** From the triangle with sides $d$ and 1, the length of the hypotenuse is: $$h = \sqrt{d^2 + 1^2} = \sqrt{(1+0.1z_1)^2 + 1}$$ Direction cosines for $F_1$: $$\cos \alpha_1 = -\frac{d}{h}, \quad \cos \beta_1 = 0, \quad \cos \gamma_1 = \frac{1}{h}$$ So components: $$F_{1x} = F_1 \cos \alpha_1 = (75+z_1) \left(-\frac{d}{h}\right)$$ $$F_{1y} = 0$$ $$F_{1z} = F_1 \cos \gamma_1 = (75+z_1) \frac{1}{h}$$ 6. **Step 3: Calculate components of $F_2$.** Given $\theta_2$ is the angle with the x-axis in the xy-plane, and $F_2$ lies in the xy-plane pointing down-right: $$F_{2x} = F_2 \cos \theta_2 = (55+z_2) \cos(60+z_4)$$ $$F_{2y} = F_2 \sin \theta_2 = (55+z_2) \sin(60+z_4)$$ $$F_{2z} = 0$$ 7. **Step 4: Find resultant components:** $$R_x = F_{1x} + F_{2x}$$ $$R_y = F_{1y} + F_{2y} = 0 + F_{2y}$$ $$R_z = F_{1z} + F_{2z} = F_{1z} + 0$$ 8. **Step 5: Calculate magnitude of resultant force:** $$R = \sqrt{R_x^2 + R_y^2 + R_z^2}$$ 9. **Step 6: Calculate coordinate direction angles of $\vec{R}$:** $$\alpha = \cos^{-1} \left( \frac{R_x}{R} \right)$$ $$\beta = \cos^{-1} \left( \frac{R_y}{R} \right)$$ $$\gamma = \cos^{-1} \left( \frac{R_z}{R} \right)$$ 10. **Summary:** - Compute $d$ and $h$ from $z_1$. - Compute $F_1$ components using $d$, $h$, and $F_1$. - Compute $F_2$ components using $F_2$ and $\theta_2$. - Sum components to get $R_x, R_y, R_z$. - Calculate magnitude $R$. - Calculate direction angles $\alpha, \beta, \gamma$. This completes the method to find the magnitude and coordinate direction angles of the resultant force.