Question Video: Finding the Moment of a Couple Equivalent to a System of Three Forces Acting on a Triangle Mathematics

Video Transcript

𝐴𝐵𝐶 is a triangle, where 𝐵𝐶 equals 48 centimeters and three forces of magnitudes 13, 13, and 24 newtons are acting along lines 𝐶𝐴, 𝐴𝐵, and 𝐵𝐶, respectively. If the system of forces is equivalent to a couple, determine the magnitude of its moment.

Let’s begin by sketching out our triangle. We have our triangle 𝐴𝐵𝐶, where the length 𝐵𝐶 is 48 centimeters. We’re going to add the forces of magnitude 13, 13, and 24 newtons to the diagram. A 13-newton force acts along the line 𝐶𝐴, as shown. Another 13-newton force acts along the line 𝐴𝐵. And then our 24-newton acts from 𝐵 to 𝐶.

Now, in fact, we can use the relationship between the length of line segment 𝐵𝐶 and the force that acts along this line to calculate the length of lines 𝐴𝐵 and 𝐴𝐶. We might spot that the ratio between 48 and 24 is one-half. And so line segments 𝐴𝐵 and 𝐴𝐶 will need to be 26 centimeters each for the ratio between the lengths and the forces in these sides to be the same.

And this is really useful because we’re told that the system of forces is equivalent to a couple. And so if we take the sum of the moments about any point on this triangle, we’ll get the same value. So we’ll determine the magnitude of the moment by taking the moments about a given point. We could choose any point on our triangle. Let’s take point 𝐴 and take the clockwise direction to be positive.

The moment, of course, is the product of the force and the perpendicular distance from the pivot. Now the only force that this applies to really is the force acting from 𝐵 to 𝐶. We see this acts at a point 26 centimeters away from 𝐴, but the 24-newton force does not necessarily act at an angle of 90 degrees. So we need to calculate the component of this force that’s perpendicular to line segment 𝐴𝐵.

To do so, we’ll begin by using the law of cosines to find the angle 𝐴𝐵𝐶 that we’ve labeled 𝜃. The law of cosines tells us that cos of 𝐵 is equal to 𝑎 squared plus 𝑐 squared minus 𝑏 squared over two 𝑎𝑐. In this case then, cos of 𝜃 is 48 squared plus 26 squared minus 26 squared over two times 48 times 26. And this means we can evaluate cos of 𝜃 exactly. 26 squared minus 26 squared is zero. Then we can simplify by dividing through by 48.

So we’re left with 48 over two times 26. And then we can divide the numerator and denominator of this fraction by four. And so cos of 𝜃 simplifies to 12 over 13. And whilst we could take the inverse cos or arccos of both sides of this equation to find the value of 𝜃, we can actually use the Pythagorean theorem to find exact values for sin of 𝜃 as well. Since the cosine ratio links the adjacent and hypotenuse in a right triangle and since five squared plus 12 squared equals 13 squared, we know that the third side in this triangle, the opposite side, must be five units. And so in this triangle, sin of 𝜃 is five over 13.

Now the reason that’s going to be useful will become apparent later on. For now, we need to work out the component of the 24-newton force that is perpendicular to line segment 𝐴𝐵. Since they are perpendicular, they meet at an angle of 90 degrees. And so the angle on the outside of the triangle here we can label as being 90 minus 𝜃. And so the component of our 24-newton force that’s perpendicular to 𝐴𝐵 is found by considering the adjacent side in this right triangle. And so it’s 24 cos of 90 minus 𝜃. This means that the moment is the product of 24 cos of 90 minus 𝜃 and the distance from the pivot, 26.

And now we notice that we can evaluate cos of 90 minus 𝜃 using the identity sin 𝜃. And of course, we worked out that sin 𝜃 was five over 13. And so the moment is going to be 24 times five over 13 times 26. Let’s simplify this expression by dividing through by 13. And we get 24 times five over one times two. But of course, five over one times two is 10, so this becomes 24 times 10, which is 240. And that is the magnitude of our moment. Since we’re working in newtons and centimeters, the moment is given in newton centimeters. And so given the information about our system of forces, the magnitude of its moment is 240 newton centimeters.