Question Video: Finding the Components of a Force from the Sum of Moments Mathematics

𝐅₁ = 𝐢 + 𝐣 and 𝐅₂ = 𝑚𝐢 − 𝐣, where 𝐅₁ and 𝐅₂ are two forces acting on the points 𝐴(2, 0) and 𝐵(0, 0) respectively. If the sum of the moments about he point of origin is 0𝐤, determine the value of 𝑚.

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Video Transcript

𝐅 one equals 𝐢 plus 𝐣 and 𝐅 two equals 𝑚𝐢 minus 𝐣, where 𝐅 one and 𝐅 two are two forces acting on the points 𝐴 two, zero and 𝐵 zero, two, respectively. If the sum of the moments about the point of origin is zero 𝐤, 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 case, we have the force 𝐅 one equals 𝐢 plus 𝐣 acting from the point 𝐴 two, zero and the force 𝐅 two equals 𝑚𝐢 minus 𝐣 acting from the point 𝐵 zero, two. The vectors 𝐫 one and 𝐫 two from the origin to the point of action of the forces are two, zero and zero, two, respectively. The question tells us that the sum of the moments of these forces about the point of origin is zero. The sum of the two moments 𝐌 one plus 𝐌 two is equal to 𝐫 one cross 𝐅 one plus 𝐫 two cross 𝐅 two.

We know the values of everything in this equation except the 𝐢-component of 𝐅 two, 𝑚. We just need to evaluate these cross products and then rearrange for 𝑚. The cross product of 𝐫 one and 𝐅 one is equal to the determinant of the three-by-three matrix 𝐢, 𝐣, 𝐤, two, zero, zero, one, one, zero. Both of these vectors are in the 𝑥𝑦-plane. Therefore, only the 𝐤-component of their cross product will be nonzero. Evaluating this determinant by expanding along the top row gives us simply two 𝐤.

Doing the same thing for the cross product of 𝐫 two and 𝐅 two gives us the determinant of the three-by-three matrix 𝐢, 𝐣, 𝐤, zero, two, zero, 𝑚, negative one, zero. Again, both vectors are in the 𝑥𝑦-plane, so only the 𝐤-component of their cross product will be nonzero. And evaluating this determinant gives us just negative two 𝑚𝐤. Therefore, the sum of the two moments 𝐌 one plus 𝐌 two is equal to two 𝐤 minus two 𝑚𝐤.

The question tells us that this sum is equal to zero 𝐤. Equating the 𝐤-component on both sides gives us two minus two 𝑚 equals zero. And solving for 𝑚 gives us our final answer: 𝑚 is equal to one.

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