Two parallel forces have magnitudes
of 10 newtons and 20 newtons. The distance between their lines of
action is 30 centimeters. If the two forces are acting in the
same direction, find their resultant 𝑅 and the distance 𝑥 between its line of
action and point 𝐴.
So what I’ve done to just start
with, I’ve actually labelled our forces. We have force 𝐴 and force 𝐵. Force 𝐴 is 10 newtons and Force 𝐵
is equal to 20 newtons. And as we’re given forces are like
and parallel, what we can say is that the resultant force is gonna be equal to force
𝐴 plus force 𝐵, which is gonna be equal to 10 plus 20. So therefore, what we can say is
that the magnitude of force are is gonna be equal to 30 N, so 30 newtons.
Okay, great! So that’s the resultant magnitude
found. What we need to do now is find the
distance 𝑥 between its line of action and point 𝐴. So in order to actually find out
what 𝑥 is, what we need to do is actually take anticlockwise and clockwise moments
about the point 𝐶. And that point 𝐶 is what I’ve
labelled where the resultant is going to act. And it’s actually the distance 𝑥
away from 𝐴.
Well, if we take a look at the
anticlockwise moment first, well the anticlockwise moment is going to be equal to
𝐹𝐵, cause that’s our force at 𝐵, multiplied by 30 minus 𝑥. And the reason it’s actually 30
minus 𝑥 is because 30 is the distance between 𝐴𝐵, and we know that the distance
between 𝐴𝐶 is actually 𝑥. So therefore, we know that the
anticlockwise moment is gonna be 20 multiplied by 30 minus 𝑥.
Okay, great! So now let’s look at the clockwise
moment about the point 𝐶. Well, the clockwise moment is gonna
be equal to 𝐹𝐴, so the force at 𝐴, multiplied by 𝑥. So therefore, our clockwise moment
about the point 𝐶 is gonna be equal to 10𝑥.
So great! We’ve actually worked out the
anticlockwise and the clockwise moments. But what we can do is actually
equate these because these are actually gonna be equal to each other. So therefore, what we actually get
is that 20 multiplied by 30 minus 𝑥 is equal to 10𝑥. So therefore, if we actually expand
the parentheses, we get 20 multiplied by 30, which gives us 600, and 20 multiplied
by negative 𝑥, which gives us negative 20𝑥. And this is all equal to 10𝑥.
And then the next stage is to
actually add 20𝑥 to each side. So we actually get that 600 is
equal to 30𝑥. So therefore, what we do is then
divide each side by 30. So we get 600 over 30 is equal to
𝑥. So we get 𝑥 is equal to 20. So therefore, what we can say is
that if we have two parallel forces that have magnitudes of 10 newtons and 20
newtons and they have a distance between their lines of action of 30 centimeters and
they act in the same direction, the resultant is gonna be equal to 30 newtons and
the distance 𝑥 between the line of action and actually the point 𝐴 where we have
the force of 10 newtons acting is going to be equal to 20 centimeters.