51. Two forces act an angle of \( 120^{\circ} \). If the greater force is 50 kg and their resultant is perpendicular to the smaller force, the smaller force is A. 20 kg B. 25 kg C. 30 kg D. 35 kg

To find the smaller force, we can use the law of cosines. Given that the forces form an angle of \( 120^{\circ} \) and the resultant is perpendicular to the smaller force, we can apply the relationship involving vector addition of forces. Using the formula: \[ R^2 = F_1^2 + F_2^2 + 2F_1F_2 \cos(120^{\circ}) \] and knowing that \( R \) is equal to the smaller force \( F_2 \) when it’s perpendicular, we can set up our equations to solve for \( F_2 \). After calculations, we find \( F_2 = 25 \) kg. Thus, the answer is B. 25 kg! Remember, always make sure to keep track of the angle’s cosine values when dealing with forces at angles to avoid any slip-ups in your calculations!