Two parallel forces have magnitudes of 24 newtons and 60 newtons as shown in the figure. The distance between their lines of action is 90 centimetres. Given that the two forces are acting in opposite directions, determine the resultant 𝑅 and the distance 𝑥 between its line of action and point 𝐴.
We’re told in the problem statement and shown in the diagram the magnitude of the two forces involved. We’ll call the first force 𝐹 sub 𝐴 and the second 𝐹 sub 𝐵. Knowing that the distance between 𝐴 and 𝐵 is 90 centimetres, we want to determine two things. We want to calculate 𝑅, the resultant of the forces 𝐹 sub 𝐴 and 𝐹 sub 𝐵, and we also want to calculate 𝑥, where 𝑥 is the distance from point 𝐴 to the line of action of the resultant force 𝑅.
Let’s start out by calculating the resultant 𝑅. As we look at our diagram, we see that positive force direction has been defined as up. So we can write that 𝑅 equals 𝐹 sub 𝐴 minus 𝐹 sub 𝐵. Entering in the values of those two forces, we find that 𝑅 is equal to negative 36 newtons. That’s the resultant force of the two original forces.
Now, we want to move on to calculating 𝑥. Given the relative magnitudes of 𝐹 sub 𝐴 and 𝐹 sub 𝐵, we can expect the point 𝑥 to exist out on the line to the right of the segment 𝐴𝐵, where 𝑥 is measured from point 𝐴. If 𝑥 identifies the line of action of our resultant force 𝑅, then we can write 𝐹 sub 𝐴 times 𝑥 minus 𝐹 sub 𝐵 times 𝑥 minus 90 centimetres is equal to zero.
We can group the terms involving 𝑥 on the left-hand side of our equation, subtract 𝐹 sub 𝐵 times 90 centimetres from both sides, and then divide both sides by 𝐹 sub 𝐴 minus 𝐹 sub 𝐵, cancelling that term on the left. We know the values for 𝐹 sub 𝐴 and 𝐹 sub 𝐵 and can plug those in now. When we calculate this fraction, we find that 𝑥 is equal to 150 centimetres. That’s the distance between point 𝐴 and the line of action of the resultant force 𝑅.