A force of 31 newtons is acting on a point 𝐴, while a parallel force of 𝐹 newtons is acting on a point 𝐵. The magnitude of the resultant of these two forces is 73 newtons. If the 31-newton force and the resultant are acting in opposite directions, what is the value of 𝐹?
Let’s begin by sketching this out. There’s a force of 31 newtons acting on a point 𝐴. Now, we don’t know the direction in which it acts. So we’ll model it as acting to the right as shown. Next, we have a parallel force of 𝐹 newtons acting on a point 𝐵. But we don’t actually know the direction in which this acts. It could act to the right as the force acting on point 𝐴, or it could still be parallel but act in the left direction.
Now, in fact, we can make a logical decision as to which direction this is acting. We’re told that the force acting on 𝐴, the 31-newton force, and then the resultant of our two forces act in opposite directions. Well, the resultant of our two forces is the sum. So for the resultant of our two forces to act towards the left, the force acting on 𝐵, the 𝐹 newton force, must also be acting in the left direction. So we can now actually find an expression that describes the resultant of these two forces.
If we take right to be the positive direction, then the sum of the forces is 31 minus 𝐹. And we subtract the 𝐹 because that’s acting in the negative direction. Then, since the magnitude of the resultant of the forces is 73 and the resultant acts towards the left in the negative direction, we can say that the sum of our forces must be equal to negative 73. That is, 31 minus 𝐹 equals negative 73. We now have a single equation in terms of 𝐹 which we can solve. We begin by adding 𝐹 to both sides, so 31 equals negative 73 plus 𝐹. Then we add 73 to both sides. So 𝐹 is equal to 31 plus 73 which is 104. So 𝐹 is 104 newtons.
It’s worth noting at this point that we can also check our answer. We can go back to our earlier step and find the resultant. Now we have the value of each of our forces. The resultant is sum of our forces is 31 minus 104 which is equal to negative 73. This means our resultant is acting in the left direction with a magnitude of 73 newtons as we expected. 𝐹 is 104 newtons.