Video Transcript
A body has a force of 10 newtons acting on it horizontally, 25 newtons acting on it vertically upward, and five newtons acting on it at an angle of 45 degrees to the horizontal as shown in the figure. What is the magnitude of the single resultant force acting on the body, and at what angle to the horizontal does it act? Give your answers correct to one decimal place.
Let’s begin by labeling our three forces 𝐅 one, 𝐅 two, and 𝐅 three. Notice how I’ve labeled these as vector forces since they’re acting in two directions. The first force, 𝐅 one, is 10 newtons acting in the positive 𝑥-direction. We can therefore say its vector is 10𝐢. The second force, 𝐅 sub two, is 25 newtons acting in the positive 𝑦-direction. So, we’ll call that 25𝐣. But what about 𝐅 three? We know its magnitude is five newtons and it’s acting at an angle of 45 degrees to the horizontal. We’re going to split this into its horizontal and vertical components. Notice by dropping these two lines in, we’ve created a right-angled triangle. And there are two ways to find the lengths of the missing sides in this triangle. We could use right-angled trigonometry.
Alternatively, if we spot that the third angle in this triangle is also 45 degrees, we see we have an isosceles triangle. And so, we know that the lengths of the shorter two sides will both be the same. And so, we use the Pythagorean theorem to find the value of 𝑥. The Pythagorean theorem tells us that the sum of the squares of the smaller two sides in the right-angled triangle is equal to the square of the hypotenuse. So here, 𝑥 squared plus 𝑥 squared is equal to five squared. This equation simplifies to two 𝑥 squared equals 25, and then we divide through by two. So 𝑥 squared is equal to 25 over two. By finding the square root of both sides of our equation, we find 𝑥 is equal to five root two over two.
Notice we didn’t worry about taking both the positive and negative square root of 25 over two. At the moment, we’re just considering this as the lengths of a triangle. We’ll look at their signs in a moment. And so, we found the horizontal and vertical components for the third force. In fact, these are both acting in the positive direction for both the 𝐢- and 𝐣-components, respectively. So, the third force is five root two over two 𝐢 plus five root two over two 𝐣.
We’re looking to find the magnitude of the single resultant force acting on the body. And so, we recall that the resultant force is the vector sum of all three forces. So, it’s 𝐅 one plus 𝐅 two plus 𝐅 three. Here, that’s 10𝐢 plus 25𝐣 plus five root two over two 𝐢 plus five root two over two 𝐣. We collect together our horizontal and vertical components. And we see that the resultant force is 10 plus five root two over two 𝐢 plus 25 plus five root two over two 𝐣.
We want to find the magnitude. Now, the magnitude of a vector is simply its length. Let’s clear some space and draw a sketch of our resultant force. By splitting it into its horizontal and vertical components, we see that the magnitude of our force can be found again by using the Pythagorean theorem. It’s the square root of 10 plus five root two over two squared plus 25 plus five root two over two squared. That gives us 31.583 and so on. Correct to one decimal place, that’s 31.6. Now, we’re working in newtons, so the magnitude of our resultant is 31.6 newtons.
We’re not quite finished, though. We want to know what angle this makes with the horizontal. Let’s label that 𝜃. This time, we see that we can use right-angled trigonometry to work out the value of 𝜃. We know the opposite and adjacent, so we’re going to use the tan ratio. Tan 𝜃 is equal to 25 plus five root two over two divided by 10 plus five root two over two. That’s 2.108 and so on. We find the value of 𝜃 by finding the inverse or arctan of this value. That’s 64.62 and so on, which, correct to one decimal place, is 64.6 degrees. We see from our sketch that this is indeed the angle that our resultant makes with the horizontal. And so, we’re finished. The magnitude of the resultant force is 31.6 newtons. And the angle it makes with the horizontal is 64.6 degrees.
