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The Addition and Resolution of Vectors: The Force Table 6J Laboratory ReportNote: Attach gphical analyses t0 Laboratory Rcpon.DATA TABLE Purpose: To analyze results...

Question

The Addition and Resolution of Vectors: The Force Table 6J Laboratory ReportNote: Attach gphical analyses t0 Laboratory Rcpon.DATA TABLE Purpose: To analyze results of different methods = vector addition ,Resultant = (magnitude and direction) Grphical Analytical" ExperimentalForces(0.2O0)g N, 0 1 (0.200) N, 01 120"Vector addition [Vector addition II(0.200)R N, 8, (0.1SO)g N, 01 (0.200)8 N (0.1S0)8 N, 0 ,Vector addition IIIVector resolution(0.300)R N, 0(0.1OO) N,0, (0.200)8 N,0, (0.300)

The Addition and Resolution of Vectors: The Force Table 6J Laboratory Report Note: Attach gphical analyses t0 Laboratory Rcpon. DATA TABLE Purpose: To analyze results of different methods = vector addition , Resultant = (magnitude and direction) Grphical Analytical" Experimental Forces (0.2O0)g N, 0 1 (0.200) N, 01 120" Vector addition [ Vector addition II (0.200)R N, 8, (0.1SO)g N, 01 (0.200)8 N (0.1S0)8 N, 0 , Vector addition III Vector resolution (0.300)R N, 0 (0.1OO) N,0, (0.200)8 N,0, (0.300) N, 8, 225" Vector addition IV Vector addition Ashon analytical calculations below:



Answers

(a) Find the magnitudes of the forces $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ that add to give the total force $\mathbf{F}_{\text { tot }}$ shown in Figure $4.35 .$ This may be done either graphically or by using trigonometry. (b) Show graphically that the same total force is obtained independent of the order of addition of $\mathbf{F}_{1}$ and $\mathbf{F}_{2} .(\mathrm{c})$ Find the direction and magnitude of some other pair of vectors that add to give $\mathbf{F}_{\text { tot. Draw these to scale on the same drawing }}$ used in part (b) or a similar picture.

In this question, we're asked to explain the difference between adding vectors and resolving a vector into its components. So toe on vectors, we take two separate vectors. So let's say A and B, and we add them together using the tip to tail method so we could start with a drop E, and then we could draw in the result since so this would give a new vector, which is a plus B. So in particular when we're adding vectors, were taking to back to separate doctors and do being some sort of process by which we only get a single vector out of the the addition. Conversely, if we are doing vector resolution so resolving vectors into its components, then if we we start with a single vector. So let's say Spector see, and we would find the components of that vector that point completely in the extraction and completely in the Y direction. So when you are resolving a vector, you're taking a single vector and you are finding its individual components. So you take a single entity and you actually end up with two numbers to additional numbers to describe that vector so you can see these processes are quite different from one another.

So again, drawing the diagram and figuring out what it means is, ah, most of the battle here. Oops, Why z x and we need to continue the Z axis downward. Um, were given F one, which appears to have no X component. Well, that's going to be convenient. So it's a 345 triangle, but it's got no X. Component. Looks wasn't drawn nicely. There we go. Okay. F one F two has all three components, but the angles are directly given, which I think so. There's no way I'm putting arrows in the ends of these, I guess, because the book does, and the angle with why is 120. So, yeah, they're all directly given. So let's go ahead and do F one, um, F one in the X direction is zero F one in the Y. Direction is well, it's going in the positive Y direction. 3/5 of 450 Well, 45 divided by five is nine nine times three is 27. So that's gonna be 270. Newton's in the positive direction. F one Z is going to be 4/5 of 450 but it is in the negative direction. So that's going to give us negative 360 Newtons. So let's move on to F two. This is actually all pretty straightforward. F two is just gonna be the 5 25 times the co sign of Ah, the X direction is going to be Alfa, which is 45 F two. Why is 5 25 who sign of the Y direction one 120 degrees and F two Z is just going to be 5 25 um co sign of Z, which is 60 degrees. So this why one should end up giving us a negative number. The other ones should be positive. So let's put them into a calculator. 5 25 Who sign of 45 is me 3 71.2 Newtons 5 25 co sign 1 20 Negative to 62.5 Newtons and 5 25 co signed 60 is just going to give us to 62.5 Newtons case. Now we have to add them together, Um, adding the X components where there is only one X components. So that's going to be this, So I'm gonna write it in vector form 3 71.2 I f. So now I'm gonna dio 270 minus to 62.5. I don't trust myself, so I'm just going to do it to 70 minus to 62.5 7.5. I thought so. Um plus 7.5 j and then in the K direction. We've got negative 3 60 plus to 62.5. Give us. Ah, negative. 97.5. Okay. I really do mean K K direction. We need Teoh sketch the vector and find the magnitude and the angles. Well, sketching the vector is what I would do next. We've got 371 in the eye direction. Then we've got 7.5, which is almost nothing in the J direction. And then we've got a negative cater exit. So it's gonna be, like down and over there, maybe something like this. All right, so now f sub are square it 3 71.2 squared plus 7.5 squared plus negative. 97.5 squared with that in a calculator. 3 80 four Alfa 3 71.2 over 3 84 bead. Ah, and Gamma are going to be very much the same. All right. Inverse Co sign, Uh, 3 71.2 Over. That's a bar 14.8. I'll just put 7.5 in the numerator. PT 8.9 and negative. 97.5. 100. All right, degrees. Now, check the

Hi in this problem. First, we will draw the diagram. In this diagram we have given a vector. A is in negative by access direction like this. This is Victor A. And the magnitude of this factor is given, which is 8 m. And the vector B is given like this. Yeah, this is after me. And the magnitude of Victor B is also given to us, which is 15 m in length. And this angle from the positive by access is also given, which is 30 degrees. Right? So first we have to find a vector plus V. Victor, we know that we can write. A vector is equal to eat meter and direction minus Jacob like this. We can also right. Victor B is equal to 15 Scient I d. I kept plus 15 cost 30 Jacob Now in part A. We have to find a victim, plus the victim. Now this will be equal toe 15 sign 30 I kept plus 15 cost 30 minus eight. Jacob, If we solve this with the help of calculator, we can get the venue. 15, 7 30 is equal to 7.5 I kept. Plus this 15 cost 30 minus eight will be equal to around 4.99 Or we can say five. Jacob, this is the value off a vector plus V vector. Right now in part B. We have to find Victor a minus. Vector B. Victor a minus. Sector V will be equal toe. Eat minus. Jacob minus 15. Sign 30 minus 15. Cost 30. Jacob. Right. So this will be equal toe minus 7.5 ICAP minus mhm 21. Jacob, this is the value off a vector. Minus the victim. This is Jacob. Now we have to find in part C minus a minus V.

Chapter one, Section seven Problem. Twenty four For the back There's Ambien. Figure you won twenty four is I have drawn here says you scale drawing to find the magnitude and direction of a the vector sum of a plus B So it begin with that. And so the one thing we learned from the vector section is how do we add these vectors together? So we'LL begin with A When you begin the Arab actors is you first start with the beginning to the end and then you add the component of the bee vector So this could be shifted in any direction and in any translation so we could begin way starting off going down eight of the three, eh? And when we add, we continue on from here with vector V upwards. Fifteen. This isn't exactly drawn to scale the thirty two b's Corrective e. And so the way this summation works is our fine electorate. Here goes from the beginning to the end. This Sarah vector, eh? Plus being so he's in terms of a drying. This gives us an idea of what we kind of have going on, but it doesn't really tell us all the information we want to know, like, what are the angles of this and what is magnitude of this? And so that's where basic trigonometry comes into play. And as you see, we're going to need to use the law of co signs. Yes, we need to pull this out from her old trigonometry training. And when we draw the triangle in quick, is there a basic triangle? And in this case, are our side right here was a fifteen and we know the angle. Here I was thirty. That's what we want to find out is the magnitude of a plus B, which is this one. So in this case, this is going to be C here and from this triangle we can see that a that's going to corresponds to the to the to the you will be to the fifteen and we just plug in everything in the sea for responsive to the thirty degrees that is opposite of C. And so when we play this in, it's C is equal to the square root of This is this is the quantity square. So we want to find sea by itself. That's gonna be Hey, I swear less. Fifteen square minus two times, eh? Times fifteen times. Co Sign of thirty degrees. Remember, is thirty degrees not thirty. Radiance. Your calculator might start off default with radiance. Just make sure to check that. So if you would uplevel listen to our calculator, we find that see is actually a nice number. It's equal to nine here, approximately nine. They're here. And so that's great way now. I found our magnitude for a plus B, but the thing is, we don't know the direction Vectors have a direction and a magnitude and direction. So that's where the next part from trigonometry comes into play the lost science. This Ah, good relationship that can give us all these qualities. And so you try the travel again in a week. You now know that this is about nine. This is a and this is fifteen. So one of the singles thirty degrees and we want to find out the angle course this angle as it would correspond away from the X axis. We got our why, Mrs X. So we just want one of the this angle in particular to find out the direction. And so, based on the law, signs we would then use this quantity. So use these parts of the equation. So we'LL have sign of thirty degrees. This what part wants to do the sea part first it's two nine is equal to sign of this angle which was just called be for now over overnight. That corresponds to this one who fifteen? Okay, the way we solve this, he's using our Andy Dandy calculator once again to solve for being. And we do that by actually using the arc sine, which is the Empress function, to get out the value with this normally would be so I'd be the arc sine of mon players to you. Alison fifteen divided by nine. I am sign of thirty degrees. So the answer for this and being fifty six point four degrees they're here, Marie And so we basically have our magnitude, which he said it was nine. And the direction of this vector from police is this bottom part is no the D six point four and so directors funny in that direction away from this quality. If we wantto say the angle away from here, then it would just be the ninety degrees minus this. Okay, And so that is for a plus b the vectors. April's being for party, for part B. We're going to want to find it for a minus. B said the vector difference. And so first thing we have to find out is or were they realizes this is the same as Vector edition of the negative vector. And so the negative actor beaches corresponds to be corresponds to the negative. In this direction, negative B still has the same magnitude fifteen Nobbs a direction and his thirty degrees away from here. So when we actually look at it in the larger view, go down a for a and then we go fifteen and destruction at an angle thirty away from here. And since lines are one hundred eighty degrees, that means that this angle inside is one hundred fifty. The police and so travel looked like this. See, See, See, this is a longer section. And so based on what we had somebody draw the triangle right here. So we know this angle is one hundred fifty fifteen. This is eight. So we want to find this side, see right here. So we do the same thing we've literally used this exact formula or it's a square less fifteen square minus, two times a day times fifteen times. Co sign of one hundred fifty degrees. Just in case you're not familiar with this location. If they put a thought here that means times till we get for here is we get about twenty two, twenty two point three, Percy. So that gives us the length. And I only want to calculate the signs. We do the same thing with the same process of the law. Of course, I So we have twenty two point three here and this angle's one hundred fifty. And we want to find out kissing, which we'LL call V again and what people does that correspond to fifteen. Until we do the same thing with the Ark co sign for the arc sine, we find out that B is equal to nineteen. Wait six, six degrees, and so this angles nineteen degrees. And so this would be our rector a minus b. Okay, so the next tests were given, it says, use the answers to find the magnitude and direction of minus a minus. B. First thing to realize is this's there's something really nice about this. There's the minuses here, which means you could pull a minor self. And so what we get is negative. A plus B and we had calculator was actor like this last time, so it's literally just the opposite. So the man into whose is the same? It's about nine and the the angle we calculator was fifty up fifty six year, which means this is about it's fifty six above here or the opposite. Like we said, ninety minus fifty six. From here, this angle is fifty six, and so the same thing with the other one word says B minus A. We do the same process. We We notice that there's a technically a plus here and a minus here. So if we take up the negative, actually get minus B plus A. And the cool thing about actors is that the community of property holds, which means you can move these to the other side. So it gives you hey minus baby and same thing like last time where quantity we have. What's this? It's really long, Better it goes down. We sense this negative. It just goes the opposite way. We said see wass or the distance of of the this wass twenty to my dream about twenty two. And so now the angle here is the same age bows before Angel here was the same as before. This is about nineteen. And so later on in exercise one point three one, there's a different approach. And so you don't have to use all this trigger trigonometry. You just have to know the components of each of these vectors, which would be a lot nicer.


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