Video Transcript
Two forces of magnitudes six ๐น and
12๐น act at a point. The magnitude of their resultant is
nine ๐น. Find the measure of the angle
between them, giving your answer to the nearest minute.
Letโs begin by imagining what these
two forces might look like. They act at some point ๐. So letโs suppose the force of
magnitude six ๐น acts in the horizontal direction as shown. The force of magnitude 12๐น acts at
the same point, but it must act in a different direction. We might assume it acts
approximately in the direction shown. Weโre trying to find the measure of
the angle between these two forces. Letโs call that ๐. Now, in order to find the value of
๐, we need to use the fact that the magnitude of the resultant of these two forces
is nine ๐น.
Remember, the resultant of the two
forces is simply their sum. But that doesnโt mean we add six ๐น
and 12๐น. These are their magnitudes. So weโre going to instead redraw
our diagram as a triangle of forces. To do so, we begin with the six-๐น
force as before. We then model the force of
magnitude 12๐น so that it starts at the terminal point of the force of magnitude six
๐น. Since the resultant is the sum of
the forces, the resultant can be modeled as starting at the initial point of the
six-๐น force and finishing at the terminal point of the force of magnitude 12
newtons as shown.
We now notice that this is a
non-right triangle and we know the lengths, which are the magnitudes of each force
of all three sides. We can therefore find the measure
of the angle between the six-๐น force and the 12๐น force in this triangle. Letโs call that ๐ผ. We can then use information about
angles in parallel lines to find the value of ๐. So how do we find the value of
๐ผ? Well, letโs label our triangle as
shown. Since we know all three sides and
weโre trying to find an angle, we can use the law of cosines. We can write that as cos ๐ด equals
๐ squared plus ๐ squared minus ๐ squared over two ๐๐. Since capital ๐ด is angle ๐ผ, the
left-hand side is simply cos ๐ผ.
Then, we substitute everything we
know about each side in our triangle. On the right-hand side, we then get
12๐น squared plus six ๐น squared minus nine ๐น squared over two times 12๐น times six
๐น. This becomes 144๐น squared plus
36๐น squared minus 81๐น squared over 144๐น squared, which simplifies even further to
99๐น squared over 144๐น squared. And of course, ๐น is part of the
magnitude of our force, so we know it cannot be equal to zero. This means weโre able to divide
both the numerator and denominator of our fraction by ๐น squared. So the cosine of our angle ๐ผ must
be equal to 99 over 144.
To find the measure of angle ๐ผ
then, letโs take the inverse or arccos of both sides. The inverse cosine of 99 over 144
is 46.567 and so on. And of course, thatโs in
degrees. Now, we wonโt round this just yet
because weโre looking to find ๐, not ๐ผ. Now, if we go back to our earlier
diagrams and compare them both, we see that angle ๐ and angle ๐ผ can be linked by a
pair of parallel lines. In fact, theyโre supplementary
angles. This means that ๐ plus ๐ผ must be
equal to 180 degrees. So ๐ is 180 minus ๐ผ. And of course, weโve calculated
๐ผ. So ๐ must be equal to 180 minus
that exact value of 46.567 and so on. That gives us 133.432.
Now the question asks us to give
our answer to the nearest minute. To do so, we need to extract the
decimal part of our answer and multiply it by 60. 0.432 and so on multiplied by 60 is
25.95. Now, of course, correct to the
nearest whole number, that will be 26. And so we found the measure of the
angle between our two forces. Correct to the nearest minute, itโs
133 degrees and 26 minutes.
