Video Transcript
Forces of magnitude 𝐅, 16, 𝐊, 18,
nine root three newtons act at a point in the directions shown on the diagram. Their resultant, 𝑅, has a
magnitude of 20 newtons. Find the values of 𝐅 and 𝐊.
We will begin by finding the sum of
the forces in the 𝑥- and 𝑦-directions, denoted 𝑅 sub 𝑥 and 𝑅 sub 𝑦, where the
positive 𝑥-direction is to the right and the positive 𝑦-direction is up. Four of the forces act in either
the 𝑥- or 𝑦-direction. The force 𝐅 acts in the positive
𝑥-direction. The 18-newton force acts in the
negative 𝑥-direction. The force 𝐊 acts in the positive
𝑦-direction. And finally, the nine root three
newton force acts in the negative 𝑦-direction. The 16-newton force acts in neither
the horizontal nor vertical direction. By creating a right triangle as
shown, we will be able to find its horizontal and vertical components. Both of these act in the positive
direction, and we will call them 𝑥 and 𝑦.
By considering the 30-degree angle
given in the diagram, we can use our knowledge of right angle trigonometry. We know that the sin of any angle
𝜃 is equal to the opposite over the hypotenuse. This means that from our diagram,
the sin of 30 degrees is equal to 𝑥 over 16. We know that the sin of 30 degrees
is one-half. And multiplying through by 16, we
have 𝑥 is equal to eight. The horizontal component of the
16-newton force is eight newtons.
We also know that the cos of any
angle 𝜃 is equal to the adjacent over the hypotenuse. This means that the cos of 30
degrees is equal to 𝑦 over 16. And as the cos of 30 degrees is
root three over two, 𝑦 is equal to eight root three. The vertical component of the
16-newton force is positive eight root three.
We now have expressions for 𝑅 sub
𝑥 and 𝑅 sub 𝑦. These can be simplified so that 𝑅
sub 𝑥 is equal to 𝐅 minus 10 and 𝑅 sub 𝑦 is equal to 𝐊 minus root three. At this stage, we don’t have enough
information to calculate the values of 𝐅 and 𝐊. However, we are told that the
magnitude of the resultant is 20 newtons, and this acts at an angle at 30 degrees to
the positive 𝑥-axis. We can therefore calculate the
values of 𝑅 sub 𝑥 and 𝑅 sub 𝑦 directly from the diagram.
Once again, we have a right
triangle. 𝑅 sub 𝑦 is equal to 20 multiplied
by the sin of 30 degrees. This means that 𝑅 sub 𝑦 is equal
to 20 multiplied by one-half. This is equal to 10 newtons. Likewise, 𝑅 sub 𝑥 is equal to 20
multiplied by the cos of 30 degrees. This is equal to 20 multiplied by
root three over two, which is 10 root three.
Substituting these values into our
equations, we have 10 root three is equal to 𝐅 minus 10 and 10 is equal to 𝐊 minus
root three. We can then solve these equations
to find the values of 𝐅 and 𝐊. 𝐅 is equal to 10 root three plus
10, and 𝐊 is equal to 10 plus root three. In order for the five forces to
have a resultant of magnitude 20 newtons as shown, 𝐅 is equal to 10 root three plus
10 newtons and 𝐊 is equal to 10 plus root three newtons.
