Video Transcript
𝐴𝐵𝐶𝐷 is a square of side length
90 centimeters. Forces of magnitudes of 25, 80, 25,
and 80 newtons are acting along the square’s sides, and two forces, each of
magnitude 45 root two newtons, are acting at 𝐴 and 𝐶 as shown in the figure. Calculate the moment of the couple
which is equivalent to this system, giving your answer in newton-centimeters.
The diagram shows three pairs of
force vectors that form force couples. The three pairs of forces are
parallel and opposite, lie on distinct lines of action, and have equal
magnitudes. As there are no other forces acting
on the system, we know that the net force is equal to zero, and the system of forces
is therefore equivalent to a couple. It is the moment of this couple
that we are trying to calculate. We recall that the moment of a
couple is equal to the magnitude of the force multiplied by the perpendicular
distance between their lines of action.
In this question, we are told that
moments in a counterclockwise direction are positive. This means that the force couples
of magnitudes 80 newtons and 25 newtons will have a positive moment, whereas the
force couple with magnitude 45 root two newtons will have a negative moment. Since the square is of side length
90 centimeters, the perpendicular distance between the 80-newton forces will be 90
centimeters. The same is true of the 25-newton
forces. The perpendicular distance between
the 45-root-two-newton forces will be equal to the diagonal 𝐴𝐶 in our square.
We can calculate this length using
the Pythagorean theorem. 𝑥 squared is equal to 90 squared
plus 90 squared. This means that 𝑥 squared is equal
to 16,200. Square rooting both sides and since
𝑥 must be positive, we have 𝑥 is equal to 90 root two centimeters. We are now in a position to take
moments for each force couple. The 80- newton force couple will
have a moment equal to 80 newtons multiplied by 90 centimeters. The 25-newton force couple will
have a moment equal to 25 newtons multiplied by 90 centimeters. Finally, we have 45 root two
newtons multiplied by 90 root two centimeters.
Recalling that the moment of this
force is acting in a clockwise direction and will therefore be negative, the moment
of the couple is equal to 80 multiplied by 90 plus 25 multiplied by 90 minus 45 root
two multiplied by 90 root two. This simplifies to 7,200 plus 2,250
minus 8,100, which is equal to 1,350.
The moment of the couple which is
equivalent to the system is 1,350 newton-centimeters.
