Video Transcript
If the force vector 𝐅 equals
negative five 𝐢 plus 𝑚𝐣 is acting at the point 𝐴 seven, three, determine the
moment of the force 𝐅 about the point 𝐵 seven, negative two.
In order to calculate the moment of
a planar force about a point, let’s recall two of the formulae we can use. If we’re considering the moment of
some force 𝐅 taken about the origin, then we calculate the cross product of the
vector 𝐫 with the vector 𝐅, where 𝐫 is the position vector of the point of
application of the force. In this case though, we want to
determine the moment of the force acting at 𝐴 about point 𝐵.
And so we need to reorient
ourselves within the coordinate plane. And to do so, we replace the vector
𝐫 with the vector 𝐁𝐀. This now is the vector moment of
the force 𝐅 acting at point 𝐴 about point 𝐵, the cross product of 𝐁𝐀 with
vector 𝐅.
So let’s begin by finding vector
𝐁𝐀. The vector 𝐁𝐀 is given by
subtracting the vector 𝐎𝐁 from the vector 𝐎𝐀. Now, of course, the point 𝐴 has
coordinates seven, three. So in the three-dimensional plane,
it has the vector seven, three, zero. Similarly, the vector 𝐎𝐁 is
seven, negative two, zero. Then, we simply subtract the
individual components. And we find the vector 𝐁𝐀 is the
vector zero, five, zero. Then, inspecting that vector force
𝐅, we see we can alternatively represent it as the vector negative five, 𝑚,
zero.
By representing each vector in this
way, we can then find the cross product of the vector 𝐁𝐀 with the vector 𝐅. Remember, the cross product can be
expressed as a determinant. If we think about the vector 𝐚
with elements 𝑎 sub one, 𝑎 sub two, 𝑎 sub three and the vector 𝐛 with elements
𝑏 sub one, 𝑏 sub two, and 𝑏 sub three, then the cross product is the determinant
of the three-by-three matrix with elements 𝐢, 𝐣, 𝐤, 𝑎 sub one, 𝑎 sub two, 𝑎
sub three, 𝑏 sub one, 𝑏 sub two, 𝑏 sub three.
So in our case, the cross product
of 𝐁𝐀 and 𝐅 is the determinant of the matrix 𝐢, 𝐣, 𝐤, zero, five, zero,
negative five, 𝑚, zero. Then, to find the determinant of
this three-by-three matrix, we multiply 𝐢 by the determinant of the two-by-two
matrix that remains if we eliminate the first row and the first column. Then, we multiply 𝐣 by the
determinant of the matrix with elements zero, zero, negative five, zero. And then we add 𝐤 times the
determinant of the final two-by-two matrix. So it’s 𝐢 times five times zero
minus zero times 𝑚 minus 𝐣 times zero times zero minus zero times negative five
plus 𝐤 times zero times 𝑚 minus five times negative five. And this simplifies really nicely
to get 25𝐤.
Now, of course, that unknown 𝑚
canceled out when we completed our cross product. And so we calculated the moment of
the force 𝐅 about our point 𝐵. It’s simply 25𝐤.
