# 252 Equivalent single force for a force and a couple

**Problem 252**

A force system consists of a clockwise couple of 480 N·m plus a 240 N force directed up to the right through the origin of X and Y axes at θ_{x} = 30°. Replace the given system by an equivalent single force and compute the intercepts its line of action with the X and Y axes.

**Solution 252**

$M_O = 480 \, \text{ N}\cdot\text{m}$

The force system above can be replaced by the following three force system shown below where the two pair into a couple which is the couple 480 N·m. The effect of these forces would be the same as the original system.

$240d = M_O$

$240d = 480$

$d = 2 \, \text{ m}$

The two opposing forces above will get canceled, thus, leaving a single force with the same effect as the three forces above which in turn has the same effect to the original system.

$(240 \sin 30^\circ)a = M_O$

$(240 \sin 30^\circ)a = 480$

$a = 4 \, \text{ m}$

$(240 \cos 30^\circ)b = M_O$

$(240 \cos 30^\circ)b = 480$

$b = 2.31 \, \text{ m}$

Thus, the equivalent single force is 240 N parallel to the given one and at a distance d = 2 m from O. The intercepts of this single force are A(-4, 0) and B(0, 2.31). *answer*