Question Video: Finding the Algebraic Measure of a Resultant Couple Mathematics

Given that the forces in the figure have magnitudes of 56, 105, and 242 newtons, find the algebraic measure of the resultant couple.

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Video Transcript

Given that the forces in the figure have magnitudes of 56, 105, and 242 newtons, find the algebraic measure of the resultant couple.

Looking at our figure, we see this pair of 56-newton forces acting in opposite directions. There’s a similar such pair of 105-newton forces and 242-newton forces. We can say then that there are three separate couples involved. Our goal is to find the measure of the resultant couple of all six of these forces. As we do this, we’ll keep in mind the sign convention that couples in the counterclockwise direction are considered positive. Therefore, couples in the clockwise direction are negative.

As we get started, we can recall that, given a couple of forces separated from an axis of rotation by a perpendicular distance 𝑑, the total moment due to that couple 𝑀 sub 𝑐 is equal to two times 𝐹 times 𝑑. Often in this relationship we’ll see a subscript indicating that the force and the distance are perpendicular to one another. We can use this general relationship for the moment due to a couple to solve for an overall moment, we’ll call it 𝑀, due to all six of our forces. That total moment is equal to the moment due to our 56-newton force couple plus the moment due to our 105-newton force couple plus that due to our 242-newton force couple.

Looking first at the moment due to our 56-newton forces, we see the lines of action of these forces are separated by 58 centimeters. The magnitude of the moment of this force couple then, leaving out units, is two times 56 times 58 over two. We also note that these forces tend to create a counterclockwise rotation, and therefore this moment is positive. Next, let’s consider the moment due to the 105-newton force couple. The lines of action of these forces are separated by 34 centimeters. So the magnitude of this couple’s moment is two times 105 times 34 over two. And we see in this case that these forces tend to create a clockwise rotation. Therefore, this moment is negative.

Lastly, for our 242-newton force couple, these lines of action are separated by 121 centimeters. The magnitude of the moment then is two times 242 times 121 over two. These forces tend to create a clockwise rotation, and therefore this moment is negative as well. We now have a complete expression for our overall moment 𝑀. Note that in each of these three terms, a factor of two and a factor of one-half multiply together to give one. Calculating this expression, we get a result of negative 29,604. In terms of units, the units of our forces are newtons, and the units of our distances are centimeters. So our final answer is that the algebraic measure of the resultant couple of these six forces is negative 29,604 newton centimeters.

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