If \( \vec{F}=2 i+3 j \) acts at point \( C \) and \( \overline{A B}=4 i+6 j \) and \( \overline{M_{A}}=\left(m^{2}+4\right) k, \overline{M_{B}}= \) \( (4 m) k \), then \( m=\cdots \cdots \) \( \begin{array}{llll}\text { (a) } 3 & \text { (b) } 2 & \text { (c) } 4 & \text { (d) } 1\end{array} \)

To find the value of \( m \), we need to analyze the forces and moments acting on points A and B. The force \( \vec{F} \) will create moments about points A and B. Given the vectors for \( \overline{M_{A}} \) and \( \overline{M_{B}} \) with dependency on \( m \), we can set up the equation based on the condition for static equilibrium, which states that the sum of the moments about any point must equal zero. Looking at the options for \( m \): - If we let \( m = 2 \) (option b), we find: - \( \overline{M_{A}} = (2^2 + 4) k = 8 k \) - \( \overline{M_{B}} = (4 \cdot 2) k = 8 k \) Both moments are equal and indicate equilibrium, suggesting \( m = 2 \) is the correct choice. Thus, the answer is: \( m = 2 \) (b).