Video Transcript
The diagram shows three coplanar forces acting at a point 𝑀. Their magnitudes are two newtons, two newtons, and eight newtons in the directions 𝑀𝐴, 𝑀𝐵, and 𝑀𝐶, respectively. Given that the measure of angle 𝐴𝑀𝐵 is 60 degrees and the measure of angle 𝐴𝑀𝐶 is 90 degrees, what is the magnitude of the resultant force? Give your answer to the nearest newton.
In order to calculate the magnitude of the resultant force, we will firstly calculate the sum of the forces in the 𝑥- and 𝑦-directions, where we’ll take 𝑀𝐴 to be the positive 𝑥-direction and 𝑀𝐶 to be the positive 𝑦-direction. The only one of our three forces not acting in one of these directions is the two-newton force in the direction 𝑀𝐵. We will therefore need to calculate the horizontal and vertical components of this force. And we can do this by creating a right triangle as shown, where the horizontal component is 𝑥 and the vertical component is 𝑦.
The magnitude of this force is two newtons. So this is the length of the hypotenuse in the right triangle. We can then use right trigonometry to calculate the values of 𝑥 and 𝑦. 𝑥 is the side adjacent to the 60-degree angle, and 𝑦 is the side opposite the angle. We know that the cos of angle 𝜃 is equal to the adjacent over the hypotenuse. This means that the cos of 60 degrees is equal to 𝑥 over two. Since the cos of 60 degrees is one-half, we have one-half is equal to 𝑥 over two. Multiplying through by two, we have 𝑥 is equal to one. The horizontal component of the two-newton force is one.
We know that the sin of angle 𝜃 is equal to the opposite over the hypotenuse. This means that the sin of 60 degrees is equal to 𝑦 over two. As the sin of 60 degrees is root three over two, we can multiply through by two such that 𝑦 is equal to root three. The vertical component of the two-newton force is root three newtons.
We now have two forces acting in the positive 𝑥-direction: two newtons and one newton. 𝑅 sub 𝑥 is therefore equal to three newtons. In the 𝑦-direction, we have an eight-newton force and a root three-newton force. 𝑅 sub 𝑦 is therefore equal to eight plus root three newtons. We can now use our knowledge of the Pythagorean theorem to calculate the magnitude of the resultant force. This is equal to the square root of 𝑅 sub 𝑥 squared plus 𝑅 sub 𝑦 squared.
In this question, the resultant force is equal to the square root of three squared plus eight plus root three squared. Typing this into the calculator, we have 10.1839 and so on. We are asked to give our answer to the nearest newton. The magnitude of the resultant force to the nearest newton is therefore equal to 10 newtons.
