𝐴𝐵 is a horizontal light rod
having a length of 60 centimeters, where two forces, each of magnitude 45 newtons,
are acting vertically at 𝐴 and 𝐵 in two opposite directions. Two other forces, each of magnitude
120 newtons, are acting in two opposite directions at points 𝐶 and 𝐷 of the rod,
where 𝐶𝐷 equals 45 centimeters. If they form a couple equivalent to
the couple formed by the first two forces, find the measure of the angle of
inclination that the second two forces make with the rod.
Looking at our diagram, we’re told
that the forces acting on points 𝐴 and 𝐵, which we can call 𝐹 sub one and
negative 𝐹 sub one, have a magnitude of 45 newtons and form a force couple. Similarly, the forces acting at
points 𝐶 and 𝐷, which we can call 𝐹 sub two and negative 𝐹 sub two, have a
magnitude of 120 newtons and also form a force couple.
If we call the moment of the first
couple 𝑀 sub one and the moment of the second couple 𝑀 sub two, we’re told that
these two couples are equivalent, or their magnitudes are equal to one another. Based on this information, we want
to find the measure of the angle of inclination that the second two forces make with
In our diagram, we’ve called this
angle 𝜃. To solve for 𝜃, we can write out
the first moment 𝑀 sub one and the second moment 𝑀 sub two and then set them equal
to one another. Since 𝐹 one acts perpendicular to
the horizontal rod, we can write that 𝑀 sub one equals the magnitude of 𝐹 sub one
multiplied by the length of segment 𝐴𝐵.
We’re told that that segment is 60
centimeters long. Therefore, 𝑀 sub one is equal to
45 newtons multiplied by 60 centimeters. The form of the equation for 𝑀 sub
two is similar, with the magnitude now of 𝐹 sub two and the length of line segment
𝐶 sub 𝐷 [𝐶𝐷]. But there’s an additional term
we’ve added in.
This sin of 𝜃 term guarantees that
the component of 𝐹 two we’re considering in this moment is perpendicular to the
distance separating the lines of action. We’re told in the exercise
statement that the distance 𝐶 to 𝐷 is 45 centimeters. So 𝑀 sub two equals 120 newtons
times 45 centimeters times the sin of 𝜃.
Since the moments 𝑀 sub one and 𝑀
sub two are equivalent, we can set 45 newtons times 60 centimeters equal to 120
newtons times 45 centimeters times the sine of the angle we want to solve for,
Dividing both sides of the equation
by 120 newtons times 45 centimeters and then taking the inverse sine of both sides,
we find that 𝜃 is equal to the arcsin of 45 newtons times 60 centimeters over 120
newtons times 45 centimeters.
When we enter that value on our
calculator, we find that 𝜃 is 30 degrees. That’s the angle our second pair of
forces makes with the line segment 𝐶𝐷.