ABC is an equilateral triangle, length of its side is 8 cm . a force of magnitude 15 N . acts along \( \overrightarrow{B C} \), then the magnitude of moment of this force about \( A \) is .....................wtoncm. \( \begin{array}{llll}\text { (a) } 40 \sqrt{3} & \text { (b) } 60 & \text { (c) } 60 \sqrt{3} & \text { (d) } 120\end{array} \)

To find the magnitude of the moment of the force about point \( A \), we can use the formula: \[ \text{Moment} = \text{Force} \times \text{Distance} \times \sin(\theta) \] Here, the force \( F = 15 \, \text{N} \) and the distance \( d \) from point \( A \) to line \( BC \) is the height of the equilateral triangle. The height can be calculated as: \[ h = \frac{\sqrt{3}}{2} \times \text{side} = \frac{\sqrt{3}}{2} \times 8 = 4\sqrt{3} \, \text{cm} \] The angle \( \theta \) between force along \( \overrightarrow{BC} \) and the line \( A \) to the line \( BC \) is \( 90^\circ \) (since we are finding the perpendicular distance): Thus, \[ \text{Moment} = 15 \times 4\sqrt{3} \times \sin(90^\circ) = 15 \times 4\sqrt{3} = 60\sqrt{3} \, \text{Ncm} \] So, the correct option is \( \text{(c) } 60 \sqrt{3} \). --- Did you know that the moment of a force is crucial in determining how objects rotate? It's essential for engineers when designing buildings, bridges, and vehicles—ensuring stability and safety. Understanding how forces impact different points lets them predict how structures will behave under load. For those curious about the elegant reasons behind equilateral triangles appearing everywhere in art and science, look into geometry’s golden ratios and symmetry. The beauty of these shapes ties closely with principles of physics, making them timeless subjects of study!