# How to find the resultant force on the object

I am so confused on how you solve these problems. One I have on my summerwork that has a 65 degrees angle then one line is 14 and the other is 9. I am sorry if this is not making any sense ;/

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#### Benit13

Math Team
This one is all about vectors.

A vector is a quantity, like force, velocity or acceleration, where you don't just specify a magnitude (how big the number is), you also specify a direction. For example, if you push something with a force, you need to know in which direction you are pushing the object. If something moves with a velocity, where is it going? Vectors represent this direction by using an arrow and the longer an arrow is, the greater the magnitude. For example, a larger velocity arrow means that the object is travelling faster.

The interesting thing about vectors is that you can split one into smaller ones or take lots of them and describe them using one vector. The first case is called "taking components", which is splitting one larger vector into 2 smaller vectors that are at right-angles to each other, and the second case is called "finding the resultant".

If you are not convinced that one vector can be the same as a set of smaller ones, consider this: if you push a cup across your desk diagonally, you can do it in two ways. You can either push it with one finger in the diagonal direction or you can push it with one finger pointing forwards and one finger pointing sideways. The cup will move in the same way in each case Now... how does the mathematics work? Basically you want to add the vectors together, but because they are in lots of directions you can't just add the magnitudes together. The trick is to split each vector you have in your problem into two vectors; a horizontal one and a vertical one. Then you can add all the horizontal ones separately and all the vertical ones separately.

Let's look at your example problem. These will be the steps you'll need to make

1. Draw a diagram (this is the most important step!)
2. Determine the horizontal and vertical components of each vector (use trigonometry for this)
2. Add up all the horizontal components of each vector
3. Add up all the vertical components of each vector
4. Make one vector out of the two components you have at the end

It will take a lot to explain each step, so I recommend you give the problem a go and, if you get stuck, make another post with your working out. I hope this helps! Thank you for taking the time to explain this to me! ok so this is what I have concluded. The magnitude is 65degress and the larger velocity is 14 and 9 would be the horizontal line. So I applied the Pythagorean theorem and came up with the resultant force on the object being 16.643. I am confused on how the magnitude has anything o do with getting the answer tho. :/ Oh wait! does 115 degrees have to do with anything? OMG....my brain ...... ****Imagine this picture and on the left side is 65 degress, the arrow closer to the top being 14units and the horizontal arrow being 9. This is what I came up with.

http://www.basic-mathematics.com/images/Obtuseangle.gif
(click search by image and its the only one that pops up) Sorry this is the only way I can explain what Ive figured out

#### Benit13

Math Team
A great start! You have the wrong answer, but don't worry.

First thing, let's improve your diagram,

i) write down the magnitude of each force (14N or 9N) by each arrow. Remember that your vectors in this problem represent force, not velocity, so you should write the units as N (for Newtons).
ii) draw a set of axes at the point the two forces originate (basically just a vertical line and a horizontal line). Label the axes x and y. Use a ruler! If you get puzzled about the horizontal vector because it overlaps, make the horizontal vector bold by making the arrow a little thicker. You should always draw the axes first.
iii) We know that the 14N force is off to one side, so we can draw a little triangle and put the 65 degree angle in the diagram. This helps a lot with the calculations later.

You should have something that looks like the attached file. I prefer to hand-draw these things but I don't have a scanner to hand Also, a couple of things to clarify before we move on

i) The magnitude is, in simple terms, how big the vector is. The magnitudes of the forces are 14N and 9N.
ii) The direction is where the arrow is pointing. In our question we are using angles or descriptions (horizontally to the right) to mark the direction.

Every vector must have a magnitude and a direction. If you don't put the direction down, you can lose marks in an exam!

That's step 1 done from my first post. Now let's look at the second step:

2. Determine the horizontal and vertical components of each vector (use trigonometry for this)
I recommend following through each calculation bit by bit on paper as you read this. It will really help, so make sure you have pencil, paper and calculator at the ready and that you have your newly improved diagram in front of you You see the triangle we drew with the angle 65 degrees? We need to calculate the lengths of the sides of that triangle. That will give us the horizontal and vertical components of the vector. We use trigonometry for this.

Well.... $$\displaystyle \sin \theta = \frac{opp.}{hyp.}$$ from SOHCAHTOA (if you're not sure what this is, look it up on Google!). In our triangle the opposite is the horizontal side, which we're after, and the hypotenuse is 14N. So, we can rearrange it a bit:

$$\displaystyle opp. = hyp. \times \sin\theta$$

and we can put the numbers in:

$$\displaystyle opp. = 14 \times \sin 65^{\circ} = 14 \times 0.9063 = 12.688$$

So the horizontal component of the 14N force is 12.688N to the left. It's important to notice that it goes to the left You now need to get the vertical component of the 14N force (hint: use cosine...)

The horizontal component of the 9N force is just 9N, because it is horizontal already The vertical component of the 9N force is 0N, because the arrow is horizontal.

Once you do this step, you're nearly done as this is the annoying bit. Also, the 9N force is a trick question. Let me know what answer you get and, if you're feeling adventurous, have a go at step 3 in my first post.

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Ahhh Thank you so much!!! However I think the 65 degrees is supposed to go to the left of the arrow. I had a tutor session and that was in the answer key but I am still confused on where to go from there

#### Benit13

Math Team
No worries, just draw the dotted line vertically down from the arrow tip instead of horizontally to make a triangle. Then perform the trigonometry on that one instead.

Quick question... could you follow through my other steps? If not I would recommend revising some trigonometry from a maths course/book. You won't be able to do this question without knowing how to use sine, cosine and tangent to get lengths of the sides of triangles.

Thank you! Okay so I got 12.688 of the 14N force even thought I inputted
12 x sin 115degrees . Which is the same you got. Now how do I figure out how to do cosine?

I have literally had to teach math to myself my whole life because I struggle with dyscalculia so I am sorry for my apparent struggle. Thank you for being so helpful!!!

#### Benit13

Math Team
No worries. I'm glad you managed to perform the sine of the force to get the horizontal component of the 14N force, but I'm worried that you don't understand why this works. The process of using cosine to get the vertical component is very easy to do once you know how.

Here's what I'm going to do: I'll provide the entire answer here in detail, but I would definitely advise you to get a maths text book and revise some trigonometry, specifically SOHCAHTOA. You should definitely do some exercises in the maths book so you are comfortable with the mathematics. These questions and many others like it will be virtually impossible to do unless you understand trigonometry. The following online resources might be able to help you out too:

SOHCAHTOA:

Finding the length of the side of a triangle:
Link: Finding a Side in a Right Angled Triangle

Here's the answer to the question you posed:

Note: You pointed out that there was a mistake in the original diagram, so I have changed it. Please confirm whether it is the right one. The change in the diagram changes the answer!

Vertical component of 14N force:
$$\displaystyle \sin \theta = \frac{opp}{hyp}$$
$$\displaystyle opp = hyp \times sin\theta = 14\sin65^{\circ} = 12.688N$$

Horizontal component of 14N force:
$$\displaystyle \cos \theta = \frac{adj}{hyp}$$
$$\displaystyle adj = hyp \times \cos\theta = 14\cos65^{\circ} = 5.917N$$

Vertical component of 9N force = 0 (it is horizontal)

Horizontal component of 9N force = 9N (it is horizontal)

The resultant force has a total vertical component of 12.688 + 0 = 12.688N
The resultant force has a total horizontal component of 9 - 5.917 = 3.083N

Use Pythagoras to get the magnitude of the resultant force:

$$\displaystyle hyp^2 = opp^2 + adj^2$$
$$\displaystyle hyp = \sqrt{12.688^2 + 3.083^2} = 13.057N$$

Use tangent to get the direction of the resultant force:

$$\displaystyle \tan\theta = \frac{opp}{adj}= \frac{12.688}{3.083} = 4.115$$
$$\displaystyle \theta = \arctan(4.115) = 76.34^{\circ}$$

So the resultant force is 13.057N, 76.34$$\displaystyle ^{\circ}$$ up from the horizontal.

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No problem 