### Video Transcript

π΄π΅πΆπ· is a square having a side
length of six centimeters. Given that the line of action of
the resultant of the forces is passing through the point πΈ, where πΈ is in the line
segment π΄π΅ and π΄πΈ equals four centimeters, determine the magnitude of πΉ.

In this question, the resultant of
the forces is nonzero. And therefore, we cannot simply
balance horizontal and vertical forces to find πΉ. What we are told is that the line
of action of the resultant force passes through the point πΈ. If the line of action of a force
passes through a point, then its perpendicular distance from that point is zero. And therefore, its moment about
that point is also zero. We therefore need to balance the
moments of all of the forces about the point πΈ and solve for πΉ. Letβs take positive moments to be
anticlockwise about the pivot point as per convention.

Cycling through the forces then,
starting with this force of two newtons from the point π΄, the line of action of
this force passes through the point πΈ. And therefore, its perpendicular
distance from πΈ is zero, and therefore its moment about πΈ is also equal to
zero. For the force of four newtons about
the point π΄, this moment will clearly have an anticlockwise turning force about
πΈ. And therefore, the moment will be
positive. The force of four newtons from the
point π· will also have an anticlockwise turning force about πΈ. So this moment will also be
positive. The force of two newtons from the
point π΅ will have a clockwise turning force about πΈ. And therefore, this will be
negative.

Assuming that πΉ is positive in the
direction indicated on the diagram, it will have an anticlockwise turning force
about πΈ. So this will also be positive. If it turns out that πΉ is in fact
negative in this direction, then this will work out in the equations since it will
have the moment of the same magnitude but opposite sign.

Now, we need to determine the
perpendicular distance from the pivot point πΈ of the line of action of all of these
forces. For the force of two newtons from
the point π΄, we have already established that this distance is zero. For the force of four newtons from
the point π΄, we are given that π΄πΈ equals four centimeters. And given that π΄π· is a horizontal
line and π΄π΅ is a vertical line, the perpendicular distance of the line of action
of this force will be four centimeters as well.

For the force of four newtons from
the point π·, we are given that the side length of the square is six
centimeters. Therefore, the perpendicular
distance of the line of action of this force from the point πΈ will be six
centimeters. For the force of two newtons from
the point π΅, we are again given that π΄πΈ is equal to four centimeters. And therefore, π΅ is equal to six
minus four, which is two centimeters. So the perpendicular distance of
the line of action of this force from the point πΈ will be two centimeters.

For the perpendicular distance of
the line of action of πΉ from the point πΈ, we need to find this length here. This gives us a right triangle
here. And we have already established
that the length of the hypotenuse π΅πΈ is two centimeters. Since the line π΅π· clearly bisects
the square, this angle in here will be 45 degrees. The length of the opposite side
will therefore be given by the length of the hypotenuse two centimeters multiplied
by the sin of 45 degrees. And this is equal to root two. Therefore, the perpendicular
distance of the line of action of the force πΉ from the point πΈ is equal to root
two.

Now, we can set the sum of all of
these moments about the point πΈ equal to zero and solve for πΉ. This gives us two times zero plus
four times four plus four times six plus two times negative two plus root two times
πΉ is equal to zero. Simplifying, this gives us 36 plus
root two πΉ is equal to zero. Subtracting 36 from both sides and
dividing by root two gives us πΉ equals negative 36 over root two. Multiplying the numerator and
denominator by root two gives us negative 36 root two over two, which gives us
negative 18 root two. We donβt need to worry about the
negative. This simply indicates that πΉ is in
the opposite direction to what is depicted on the diagram. The magnitude of πΉ is equal to 18
root two newtons.