Video Transcript
Two forces 𝐅 one and 𝐅 two are acting at the points 𝐴 four, one and 𝐵 three, negative one, respectively, where 𝐅 one equals three 𝐢 minus 𝐣 and 𝐅 two equals 𝑚𝐢 plus two 𝐣. If the sum of the moments of the forces about the origin point is zero, determine the value of 𝑚.
Recall that the moment 𝑚 of a force 𝐅 acting from a point 𝑃 about a pivot point 𝑂 is given by 𝐫 cross 𝐅, where 𝐫 is the vector 𝑂 to 𝑃. In this question, we are told that the sum of the moments of 𝐅 one and 𝐅 two about the point of origin is equal to zero or the zero vector. We’ll call the moment of 𝐅 one 𝐌 one and the moment of 𝐅 two 𝐌 two. If we let 𝐫 one equal the position vector of the point 𝐴 which 𝐅 one acts from and 𝐫 two equal the position vector of the point 𝐵 which 𝐅 two acts from, then the sum of these two moments is 𝐫 one cross 𝐅 one plus 𝐫 two cross 𝐅 two.
𝐫 one is therefore the vector four, one; 𝐅 one is the vector three, negative one; 𝐫 two is also the vector three, negative one; and 𝐅 two is the vector 𝑚, two. These two cross products are then the determinants of the three-by-three matrices 𝐢, 𝐣, 𝐤, four, one, zero, three, negative one, zero and 𝐢, 𝐣, 𝐤, three, negative one, zero, 𝑚, two, zero. All of the vectors involved are in the 𝑥𝑦-plane and have a 𝐤-component of zero. Therefore, only the 𝐤-component of their cross products will be nonzero. Taking these two determinants by expanding along their top rows gives us negative seven 𝐤 plus six plus 𝑚 𝐤. This simplifies to 𝑚 minus one 𝐤.
We are told in the question that this sum of the two moments is equal to the zero vector, so we have 𝑚 minus one 𝐤 equals the zero vector. For a vector to be equal to the zero vector, all of its components must be equal to zero. So 𝑚 minus one is equal to zero. Rearranging for 𝑚 gives us our final answer, 𝑚 equals one.
