Coplanar forces act outwards from a common point. Three forces have the following magnitudes and directions. The angles being measured clockwise from a horizontal datum, 80N,0°; 50N,60°; 100N,150°. a. Draw the force diagram b. Determine the magnitude and direction of the resultant force

The given angles are measured clockwise (CW) from a horizontal datum.

• For $F_1 = 80 \, \text{N}$ at $0^\circ$ CW: $\theta_1 = 0^\circ$ CCW.
• For $F_2 = 50 \, \text{N}$ at $60^\circ$ CW: $\theta_2 = 360^\circ - 60^\circ = 300^\circ$ CCW (or $-60^\circ$).
• For $F_3 = 100 \, \text{N}$ at $150^\circ$ CW: $\theta_3 = 360^\circ - 150^\circ = 210^\circ$ CCW (or $-150^\circ$).

For $F_1$:

$$F_{1x} = 80 \cos(0^\circ) = 80 \times 1 = 80 \, \text{N}$$ $$F_{1y} = 80 \sin(0^\circ) = 80 \times 0 = 0 \, \text{N}$$

For $F_2$:

$$F_{2x} = 50 \cos(300^\circ) = 50 \times 0.5 = 25 \, \text{N}$$ $$F_{2y} = 50 \sin(300^\circ) = 50 \times (-\frac{\sqrt{3}}{2}) \approx -43.30 \, \text{N}$$

For $F_3$:

$$F_{3x} = 100 \cos(210^\circ) = 100 \times (-\frac{\sqrt{3}}{2}) \approx -86.60 \, \text{N}$$ $$F_{3y} = 100 \sin(210^\circ) = 100 \times (-0.5) = -50 \, \text{N}$$

$$R_x = F_{1x} + F_{2x} + F_{3x} = 80 \, \text{N} + 25 \, \text{N} - 86.60 \, \text{N} = 18.40 \, \text{N}$$ $$R_y = F_{1y} + F_{2y} + F_{3y} = 0 \, \text{N} - 43.30 \, \text{N} - 50 \, \text{N} = -93.30 \, \text{N}$$

$$R = \sqrt{R_x^2 + R_y^2}$$ $$R = \sqrt{(18.40 \, \text{N})^2 + (-93.30 \, \text{N})^2}$$ $$R = \sqrt{338.56 + 8704.89}$$ $$R = \sqrt{9043.45}$$ $$R \approx 95.097 \, \text{N}$$

$$\alpha = \arctan\left(\frac{R_y}{R_x}\right)$$ $$\alpha = \arctan\left(\frac{-93.30}{18.40}\right)$$ $$\alpha \approx \arctan(-5.07065)$$ $$\alpha \approx -78.86^\circ$$

This angle is measured counter-clockwise from the positive x-axis. To express it as a clockwise angle from the horizontal datum, we take the absolute value:

$$\text{Direction} = 78.86^\circ \text{ clockwise from the horizontal datum}$$

The magnitude of the resultant force is $\boxed{\text{95.10 N}}$ and its direction is $\boxed{\text{78.86}^\circ \text{ clockwise from the horizontal datum}}$.