Video Transcript
In the figure, forces having
magnitudes of 61, 43, 100, and 𝐹 newtons are acting on the light rod, and the rod
is in equilibrium horizontally. Determine the length of the line
segment 𝐷𝐴 and the magnitude of 𝐹.
The key to answering this question
is to spot that the rod is in equilibrium. So what does that mean? Well, it means two things. Firstly, the sum of all the forces
acting on the rod is equal to zero. In this case, we’ll consider the
sum of the forces acting in a vertical direction. Secondly, it also means that the
sum of the moments of our forces is also equal to zero, where the moment is
calculated by multiplying the force acting at a point by the perpendicular distance
from the line of action of this force to the point about which the object will
turn.
So let’s begin by considering the
sum of the forces acting on our diagram. Let’s define the positive direction
to be upwards so that 𝐹 is acting in the positive direction, whilst 61, 43, and 100
are acting in the negative direction. We can therefore say that the sum
of our forces must be 𝐹 minus 61 minus 43 minus 100. And of course we know that this sum
is equal to zero.
Now, in fact, the expression on the
left-hand side simplifies to 𝐹 minus 204. So 𝐹 minus 204 is equal to
zero. We’ll solve for 𝐹 by adding 204 to
both sides. So 𝐹 is equal to 204 or 204
newtons. So we’ve now calculated the value
of 𝐹 and we’ve done everything we can with the sum of our forces. So we move on to the second
criteria. The sum of the moments of our
forces is equal to zero.
Now, we’ve been defined a positive
direction here. That’s the counterclockwise
direction. And we know that a moment is
calculated by multiplying the force by the perpendicular distance of this force from
the point about which the object is trying to rotate. We have a few distances on here,
but there is one distance that we’re missing.
Let’s define the distance between
𝐴 and 𝐷, so the length of the line segment 𝐷𝐴, which coincidently we’re trying
to find, to be 𝑥 centimeters. And then once we have that
information, we pick a point about which to calculate our moments. Now, we can pick any point on the
rod itself. We’re going to pick 𝐷 here. Now, the reason we’re going to pick
𝐷 is because the force 𝐹 is acting at this point. And had we not calculated 𝐹 yet,
we still would’ve been able to calculate the moments about this point since the
moment of 𝐹 would’ve been zero.
It’s also worth noting that we
generally choose to work in newton meters when calculating moments. But actually, our dimensions are in
centimeters. So we’re going to be working in
newton centimeters throughout. And that’s absolutely fine as long
as we’re consistent. So let’s find the moment of our
61-newton force. This force is trying to turn the
object in a counterclockwise direction. So its moment will be positive. It’s 100 centimeters away from 𝐷,
so the moment is 61 times 100.
Moving from left to right, we’re
now going to deal with the force at 𝐶. Once again, this is trying to move
the object in a counterclockwise direction. So its moment is positive. But now it’s 43 times 50. As we said, the force 𝐹 is zero
centimeters away from 𝐷. So we don’t need to worry about
that moment. And instead we move on to the force
at 𝐴.
Now, this force is trying to move
the object in a clockwise direction. So its moment is actually going to
be positive. And it’s 100 times the distance
from 𝐷, which is 𝑥. We know of course that the sum of
these moments is zero. So we can form an equation in
𝑥. This equation simplifies to 8250
minus 100𝑥 equals zero. Then we add 100𝑥 to both sides and
finally divide through by 100. So 𝑥 is 82.5. And we see that the length of line
segment 𝐷𝐴 is 82.5 centimeters.
