### Video Transcript

In the figure below, ๐น sub one and ๐น sub two are two parallel forces measured in newtons, where ๐
is their resultant. If ๐
is equal to 30 newtons, ๐ด๐ต equals 36 centimeters, and ๐ต๐ถ equals 24 centimeters, determine the magnitude of ๐น sub one and ๐น sub two.

In this question, we have two parallel coplanar forces, ๐น sub one and ๐น sub two, acting in opposite directions. We are also given the resultant force ๐
, which is equal to 30 newtons. The distance from point ๐ด to ๐ต is 36 centimeters, and the distance from ๐ต to ๐ถ is 24 centimeters. The lines of action of our forces ๐น sub one, ๐น sub two, and ๐
are not perpendicular to the line segment ๐ด๐ถ. However, as our three forces are parallel, we can add the angle ๐ to our diagram as shown.

We can calculate the perpendicular components of these forces using our knowledge of right angle trigonometry. These are equal to ๐น sub one sin ๐, ๐น sub two sin ๐, and ๐
sin ๐. These will be useful when we come to take moments, as the moment of a force is equal to the magnitude of force multiplied by the perpendicular distance to the point at which we are taking moments. Whilst we can take moments about any point on our line, in this question, we will do so about point ๐ด.

We will consider moments acting in the counterclockwise direction to be positive and those acting in the clockwise direction to be negative. This means that the moment ๐ sub one of the force ๐น sub one acts in the positive direction and is equal to ๐น sub one sin ๐ multiplied by 36. We can repeat this process for ๐ sub two, which is the moment of the force ๐น sub two. As this acts in the negative direction, this is equal to negative ๐น sub two sin ๐ multiplied by 60.

Our expressions for ๐ sub one and ๐ sub two can be simplified as shown. We know that the distance ๐ฅ from the line of action of the resultant force to the point at which we are taking moments is equal to the sum of the moments divided by ๐
. In this case, we are taking moments about the point where the resultant acts. Therefore, ๐ฅ is equal to zero. This means that the sum of our two moments must equal zero. 36 multiplied by ๐น sub one sin ๐ plus negative 60 multiplied by ๐น sub two sin ๐ equals zero. Since sin ๐ cannot be equal to zero, we can divide through by this. We can also divide through by 12 such that three ๐น sub one minus five ๐น sub two equals zero. As there are two unknowns here, we will call this equation one.

We will now consider the resultant force and the fact that this is equal to the sum of the other forces. Going back to our initial diagram, if we let the positive direction be vertically upwards, we have ๐
is equal to ๐น sub one plus negative ๐น sub two. Since ๐
is equal to 30 newtons, we have 30 is equal to ๐น sub one minus ๐น sub two. Adding ๐น sub two to both sides of this equation, we have ๐น sub one is equal to 30 plus ๐น sub two. We will call this equation two. And we now have a pair of simultaneous equations that we can solve by substitution.

One way of doing this is to substitute the expression for ๐น sub one in equation two into equation one. This gives us three multiplied by 30 plus ๐น sub two minus five ๐น sub two is equal to zero. Distributing the parentheses gives us 90 plus three ๐น sub two. The left-hand side then simplifies to 90 minus two ๐น sub two. By adding two ๐น sub two to both sides and then dividing through by two, we have ๐น sub two is equal to 45. Substituting this value back into equation two gives us ๐น sub one is equal to 30 plus 45, which is equal to 75. The magnitude of the two forces ๐น sub one and ๐น sub two are 75 newtons and 45 newtons, respectively.