Video Transcript
The given figure shows two forces
each of magnitude 267 newtons acting on two edges of a lamina in the form of a
rectangle with dimensions 𝑥 equals 42 centimeters and 𝑦 equals 84 centimeters. Find the moment of the couple if
tan 𝜃 is equal to three- quarters.
We begin by adding the dimensions
of our rectangle, 42 centimeters and 84 centimeters, to our diagram. The two forces of magnitude 267
newtons act at an angle of 𝜃 to the vertical. This means that the forces are
parallel but act in opposite directions. Whenever we have a pair of equal
and opposite forces, that pair is known as a couple. And if those two forces act at
points a perpendicular distance 𝑑 away from each other, we say that the moment of
the couple is equal to 𝐹 multiplied by 𝑑.
We can use this relationship to
solve for the moment of the couple in this question. However, in order to do this, we
will firstly calculate the horizontal and vertical components of the forces. The two vertical forces are
separated by a perpendicular distance of 84 centimeters. And the two horizontal forces are
separated by a perpendicular distance of 42 centimeters. We will call the vertical and
horizontal components 𝐹 sub 𝑣 and 𝐹 sub ℎ. We can then use our knowledge of
right angle trigonometry to calculate these. Recalling that sin 𝜃 is equal to
the opposite over the hypotenuse and cos 𝜃 is equal to the adjacent over the
hypotenuse, we have sin 𝜃 is equal to 𝐹 sub ℎ over 267 and cos 𝜃 is equal to 𝐹
sub 𝑣 over 267.
We are told that the tan of 𝜃 is
equal to three-quarters, and this is equal to the opposite over the adjacent. As our triangle is right angled, we
therefore have a Pythagorean triple with arbitrary length units of three, four, and
five, where sin 𝜃 is equal to three-fifths and cos 𝜃 is equal to four-fifths. Substituting these into our
equations and then multiplying through by 267, we have 𝐹 sub ℎ is equal to 160.2
and 𝐹 sub 𝑣 is equal to 213.6. These are the horizontal and
vertical components of the forces in newtons. Adding these forces to our diagram,
we see that we’ve replaced the original couple with two new couples.
We are now in a position to
calculate the moment of each of these. If we consider the horizontal
couple, we have a moment equal to the force of 160.2 newtons multiplied by 42
centimeters. This is equal to 6728.4 newton
centimeters. Repeating this to calculate the
moment of the vertical couple, we have 213.6 newtons multiplied by 84
centimeters. This is equal to 17942.4 newton
centimeters.
Next, we notice that the moment of
both of the couples act in a counterclockwise direction. This is the positive direction and
means that the sum of the horizontal and vertical moments will be equal to the
moment of the original couple. We need to add 6728.4 and
17942.4. This is equal to 24670.8. And we can therefore conclude that
the moment of the couple is 24670.8 newton centimeters.
