Video Transcript
The angle between two forces, 𝑎
one and 𝑎 two, is 75 degrees. Their resultant is 2,900 newtons
and makes an angle of 45 degrees with 𝑎 one. Find the forces 𝑎 one and 𝑎
two. Give your answers to two decimal
places.
Before we do anything, we’re going
to begin by drawing a force diagram. Here are our two forces 𝑎 one and
𝑎 two, which meet at an angle of 75 degrees. Then, we have this resultant force
of 2,900 newtons that makes an angle of 45 degrees with 𝑎 one. Remember, the resultant force is
just the single force that we get by combining a system of forces, so here 𝑎 one
and 𝑎 two. Our job is to find the forces 𝑎
sub one and 𝑎 sub two. And so we’re going to add a couple
of things to our diagram.
Firstly, we add in lines parallel
to those representing the forces 𝑎 one and 𝑎 two. This gives us a parallelogram. And so we know that not only are
the opposite sides parallel, but they’re also equal in length. And that’s going to be useful later
down the line. Then we use the fact that alternate
angles are equal. And this angle must be 45
degrees. Then, by subtracting 45 from 75, we
get two angles of 30 degrees. The third angle is found by
subtracting 45 and 30 from 180 degrees, since there are 180 degrees in a
triangle. And so we see that the third angle
in both our triangles is 105 degrees.
Let’s separate out our
triangles. And we see that we’re dealing with
a pair of identical non-right-angled triangles. We have a length of 2,900 in both
triangles. And then we have 𝑎 sub one
here. And since we said that opposite
sides in a parallelogram are parallel and equal in length, we have 𝑎 sub two over
here.
Since we know the measure of all
the angles in our triangle and one of its lengths, we can use the law of sines to
find the other two. Let’s label our triangle as
shown. And to begin with, we’ll work out
the value of 𝑎 sub one. This means we want 𝑎 over sin 𝐴
equals 𝑐 over sin 𝐶. Substituting everything we know
about our triangle into this formula gives us 𝑎 sub one over sin of 30 degrees
equals 2,900 over sin of 105. We’re going to multiply both sides
by sin of 30 to find the value of 𝑎 sub one. And so 𝑎 sub one is 2,900 over sin
of 105 times sin of 30, which correct to two decimal places is 1,501.15. So 𝑎 sub one is 1,501.15
newtons.
Let’s clear some space and perform
the same process to find 𝑎 sub two. This time, we’re going to use 𝑏
over sin 𝐵 equals 𝑐 over sin 𝐶. Substituting into the formula, and
we see that we can solve for 𝑎 sub two by multiplying both sides by sin of 45. And 𝑎 sub two then is 2,900 over
sin of 105 times sin of 45. And that, correct to two decimal
places, is 2,122.95. 𝑎 sub one is 1,501.15 newtons, and
𝑎 sub two is 2,122.95 newtons.
