1. **Problem Statement:** Find the algebraic measure of the moment of a force of magnitude 200 N about point O. 2. **Given Data:** - Force magnitude, $F = 200$ N - Distance from point O horizontally, $d_x = 60$ cm - Distance from point O vertically, $d_y = 40$ cm - Force direction angle below horizontal, $\theta = 30^\circ$ 3. **Formula for Moment:** The moment $M$ about point O is given by the cross product of the position vector $\vec{r}$ and the force vector $\vec{F}$: $$M = rF\sin(\alpha)$$ where $\alpha$ is the angle between the position vector and the force vector. 4. **Step 1: Find the position vector magnitude $r$** $$r = \sqrt{d_x^2 + d_y^2} = \sqrt{60^2 + 40^2} = \sqrt{3600 + 1600} = \sqrt{5200} = 72.11 \text{ cm}$$ 5. **Step 2: Find the angle $\phi$ of the position vector relative to horizontal** $$\phi = \tan^{-1}\left(\frac{d_y}{d_x}\right) = \tan^{-1}\left(\frac{40}{60}\right) = 33.69^\circ$$ 6. **Step 3: Find the angle $\alpha$ between $\vec{r}$ and $\vec{F}$** Force is $30^\circ$ below horizontal, so angle of force vector relative to horizontal is $-30^\circ$. $$\alpha = \phi - (-30^\circ) = 33.69^\circ + 30^\circ = 63.69^\circ$$ 7. **Step 4: Calculate the moment** $$M = r F \sin(\alpha) = 72.11 \times 200 \times \sin(63.69^\circ)$$ Calculate $\sin(63.69^\circ) \approx 0.894$: $$M = 72.11 \times 200 \times 0.894 = 12877.5 \text{ N.cm}$$ 8. **Step 5: Determine the sign of the moment** The force tends to rotate the beam clockwise about point O, so the moment is negative: $$M = -12877.5 \text{ N.cm}$$ 9. **Step 6: Compare with given options** Closest option is (a) $-14392.3$ N.cm, but our calculation is $-12877.5$ N.cm. Rechecking angle: If we consider the force angle relative to the position vector differently, using $\alpha = 30^\circ + 33.69^\circ = 63.69^\circ$ is correct. Alternatively, calculate moment as: $$M = F \times \text{perpendicular distance}$$ Perpendicular distance $d_{\perp} = r \sin(\alpha) = 72.11 \times 0.894 = 64.44$ cm Moment: $$M = 200 \times 64.44 = 12888 \text{ N.cm}$$ This confirms the previous result. **Final answer:** $$\boxed{-12877.5 \text{ N.cm}}$$ Since none of the options exactly match, the closest is (a) $-14392.3$ N.cm, likely due to rounding or diagram interpretation.