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.
