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UU110.0 points Lou' $ lalesL invention_ aited at uban dob oWnerS the X-R-Leash Il iS Inade rubber-like material that exerts force (-2.6 N/In) € (S0 N/m?...

Question

UU110.0 points Lou' $ lalesL invention_ aited at uban dob oWnerS the X-R-Leash Il iS Inade rubber-like material that exerts force (-2.6 N/In) € (S0 N/m?) 12 when is stretched distance The ad elaims ~You ll never g0 back youI old dos leash alter Fou had the thrill of an X-R-Leash ex- perience. And you 'Il see new look of respect in the eyes of your proud pooch: Find the work done OLL dog by the leash if the persOn remains stationary. and the dob bounds OfE. stretching the X-R-Leas

UU1 10.0 points Lou' $ lalesL invention_ aited at uban dob oWnerS the X-R-Leash Il iS Inade rubber-like material that exerts force (-2.6 N/In) € (S0 N/m?) 12 when is stretched distance The ad elaims ~You ll never g0 back youI old dos leash alter Fou had the thrill of an X-R-Leash ex- perience. And you 'Il see new look of respect in the eyes of your proud pooch: Find the work done OLL dog by the leash if the persOn remains stationary. and the dob bounds OfE. stretching the X-R-Leash [rom 0 m Lo € 12 I Answer in uils of kI



Answers

A person is walking three dogs fastened to separate leashes that meet in a connective hub, leading to a single leash that she is holding. Dog 1 applies a force $\mathrm{N} 60^{\circ} \mathrm{W}$ with a magnitude of $8,$ Dog 2 applies a force of N $45^{\circ} \mathrm{E}$ with a magnitude of $6,$ and $\mathrm{Dog} 3$ moves directly $\mathrm{N}$ with a magnitude of $12 .$ Find the magnitude and direction of the force $\mathbf{w}$ that the walker applies to the leash in order to counterbalance the total force exerted by the dogs.

It's exercise. We have a force that that is equal to 98 I plus 132 j plus 32 k our Newton's where I, J and K are the defectors in the direction of the X, y and Z axes, respectively. And in question. A. We have to find the magnitude of this force. So you know that the magnitude of the force is equal to the square root of the X component f ax squared plus F y squared. Let's FZ square. So this is equal to effect, which is 98 squared plus F y, which is 132 square plus F Z, which is 32 square 98 squared is 9604 132 square is 17,000 424 and 32 square is 1024 and this is equal to 167.5 Newtons. This is the magnitude of the force, and the answer to question a and in question be you have to calculate the angle that the force does with the vertical. This means that we have to calculate the angle that it does with the axe access. Ah, now notice. That's the force. The the dot product between the force and the I vector Ah, the factory zone A doctor in the X direction. So this is equal to the magnitude of the force times the magnitude of the eye vector, which is just one. So I won't bother writing it times the coal sine of the angle between the, uh, between the i vector and ah, the force f And this is exactly the angle we want to calculate because it's the angle between the X direction and the force. This is I This is the fourth. So this is the angle that we want to calculate. So what? I'm gonna do this right? The coal sign of F as if that I divided by the magnitude of death. Now, after I is just f x Kate's the projection of the vector acts along the developed a ref along the extraction. So this is f X divided by the magnitude of death. F X is 98 the magnitude of F has recalculated previously is 167 point five and this is equal to zero point 585 So Fada is the Ark cool sign of 0.585 which is equal to 54.2 degrees. And this is the answer to question. Be ah, and this concludes the exercise.

This question talks about a dog walker. He's walking two dogs at the same time. They're pulling in different directions, and we want to know what's the force that she has to apply to keep all the dogs in equilibrium. So we're told that one of the dogs I'm gonna drove the point where the leashes diverge and the direction of Due North, one of the dogs is pulling 60 degrees west of due North. So that's 60 degrees with a force of eight or not. Total of the units Arts just eight with the other dog is pulling with a force of six 45 degrees east of due north. So let's, um let's figure this out and see first, what is the resultant force? We're going to add these two vectors together to find what the resultant force is, and we know that with the dog Walker is gonna have to apply is just gonna be the opposite of that. So if the dog walker can counteract both dogs, then the system will remain stationary, and that's what we want. So the way we're gonna find the result in vector of these two dogs is just by adding the vectors. Now, these vectors aren't at 90 degree angles with each other, so it's gonna be a little tricky to, um, do that. What we're gonna need to do is break them down into their components and go from there. So the way I'm gonna do this is I'm gonna call, um, this the horizontal and this the vertical, and we're going to find it that way. So let's start with the vector of length. Six. So, in the horizontal direction, we know that we have, um it's gonna be like this. The angle 45 degrees. We want to find the opposite side in this triangle, right? So even though we're finding the horizontal component, we want to find the opposite side of this triangle that we've made. I'll draw a little bit nicer where the radius is six. And this interior angles 45 and we want to find the opposite side. So we're gonna be using sign, even though it's the, uh the horizontal length that we're finding. And for the vertical length here, we're gonna be since this is the adjacent side to the angle will be using co sign so the horizontal will be six times the sign of 45 degrees and the vertical will be six times the coast sign of 45 degrees is a little tricky to follow because normally we have co sign being the X and sign being the why. This time it's flipped because our angle is actually from due north and not from Duke East, which we called the horizontal. So the same sort of thing is gonna be happening with, um, the, uh, vector of length eight. But it's gonna be even trickier because now it's just further toothy left. So, actually, what will be doing is calling this'll theon visit side here that will soon be a horizontal component but will multiply it by a negative because in the other direction, So it's gonna be negative eight times sign of 60 which is our angle. And then it'll be positive eight times the co sign of 60 which is our vertical. So now that we have all our components, we can add this together to find the direction and the magnitude when we add them together and we out of our components will get the vertical component of negative 2.69 and the horizontal component of 8.24 So what we have is a angle that looks like this with negative 2.69 and 8.24 vertical. And this is our result in Fekter. We need to find the magnitude in the direction of this so that we can find the magnitude and direction that the dog walker needs to apply to keep it stationary. So the magnitude is gonna be pretty easy to find the magnitude. Of course, we know is just the square root of each component squared. And when we do that, we'll find that the magnitude is actually 8.67 and we don't have units for members, which is 8.67 now. The angle is gonna be a little bit trickier when we do what we normally do where we say theta is equal to the inverse tangent of the horizontal component divided by the vertical component 2.69 We're gonna get a an answer that is something negative. So when you plug it into our calculators, what will find its inverse tangent of 8.24 divided by negative 2.69 is negative, 71.92 So this is 71.90 degrees below the horizontal. That's of course, not what we would expect because we're expecting more than 90 degrees above the horizontal. That's because Tangent inverse tangent always has, technically two answers because it could be the numerator or the denominator. That's negative. The calculator will assume that it's the numerator that's negative so that it is has a negative rise and a positive run, which will give you unanswered in the first or fourth quadrants. We're looking for an answer in the second quarter, so that means we need to add 180 degrees to this, uh, to this angle. So stay tha You know what the calculator tells us? Its neck of 71.92 is actually in the direct opposite direction, and so it's gonna be 108 0.0 eight degrees that's degrees above the horizontal were given it in terms of cardinal directions, so we know that no worth is 90 degrees above the horizontal. So if we subtract 90 degrees from this, what we'll get is 18.8 degrees, so there's 18.8 degrees between this result of Victor and North. So we call the direction here north, 18.8 degrees west. So that's the direction of the results in Fechter. Now to turn it around, it will be pretty easy. Remember the dog Walker needs Teoh exactly opposite be exactly opposite the results in Vector to stabilize the system. So she's gonna have to pull with 8.67 units of force in the exact opposite direction, which is easy enough to find it's just south, 18.8 degrees east, and that is your final answer.

Over question 86. So talks about there are three dogs which are fastened to separate leashes, Add media to connective hub dog. Want a place for 60° north with the northwest with the magnitude of it. Doctor applies for a fair degree clockwise of north of the magnitude of six and doctor removes directly north with the magnitude of 12. We need to find the magnitude and direction of the force which the walker must apply to counterbalance the total force and by other dogs. Right? So let's draw a coordinate plane and let's represented represent these forces in the form of work. So this is the coordinate plain. Uh for a 60° northwest 63 northwest will look something like this. So this is eight, This is 60° 45 degree. Will come over here for the third degree clockwise of north. So that's going to look like this. And We have a magnitude of six. And adultery applies 12 Newtons, which is directly not. So that's somewhere over here. So we need to we will first write all the factors and and the I. N. G. A. For mark. So for the for the W. One which is the avon with eight newtons, that's gonna be the computer X. Company is going to be towards negative. So that's gonna be negative. Eight sign 60 I And towards the wire component. That's eight costs 60 and you for W. Two. And we can simplify this as well. Since 19 sixties route three or so that's gonna be negative for three I plus four G. Likewise the W. Two is going to be six cost 45 degree cost 45 schools 45 degree J. Cap And six signed 45 degree towards the icap. So that's going to be six over route to I plus six over root two cheap And third workers. Just 12 J. Because it destroys the positive Y axis. So the net force W. Is going to be the some network actor due to these these dogs. Let's call it W. Four. That will be the sum of all these three Vectors. That's going to be we have to learn I with I. So that's gonna be negative for room three. All right. Plus six over route to I. And then we have to R. J. B. J. So they're 12 and three. So four plus 12 is 16. That's gonna be 16 plus six over root two. Check So the magnitude of this will be same as the magnitude which the Walker must apply. So that's going to be square root of -4. Route three plus six over root two square plus 16 plus six over root two whole square. So this is going to be equal to this is going to be equal to, We're going to take the help of calculator over here, that's going to be six over root two minus four, treat square. And plus we have 16 plus. This can be written as a six hour route to. Uh So this is gonna be we take the square out of this, so that's going to be 20 point, it's going to be the magnitude is coming up 20 point for two newton's. And when we talk about when we talk about the direction, so let's first talk about the direction of W. Four. So for the direction of the W four, they can't heat is going to be the Y. Component over the X. Number and that's going to be 16 plus six over root two over negative four route three plus six over route. So uh let me grab my calculator again over here. Let's find this value. So it's going to be minus four. Cream 16 plus Cool, We're on so town and we're so so that's coming in the negative direction. That's coming as -8244. What the negative means is although the resultant vector will be somewhat somewhere over here. So the angle which this which the stand, it is giving us in the negative sense. But the actual angle which the dogs are applying would be would be over here. So, since the dogs are applying an angle over here, so the Walker must apply an angle which is exactly opposite to this one. So which should be this. So actually this would be that angle with the Walker is applying. So we're going to say that the Walker must apply 82. The four star ready to .44° south of East. And that's where the walker must apply. The force. Thank you

Question radio faces. That is based on the concept of resultant force. A person is walking two dogs fastened to separate leashes that meet in a connective hub Leading to a single leash that she's working. The one applies a force of 30° northwest, the magnitude of eight, and doctor applies a force of 45 degree clockwise of north, with a magnitude of six. So find the magnitude and direction of the of the first W that the walker applies to leash to the leash in order to counterbalance the total force exerted by their dogs. All right, so, we need to find resultant force of about the dogs first, and then we need to find exactly opposite force to guard which the walker will apply. All right, so let's uh let's first try to draw these vectors. So this is the coordinate plain struck about dog one, which is represented by Let's call it W one. So it applies a 60° northwest. So this is 60° northwest, call it of magnitude eight. And the doctor applies 45 degree clockwise. So this is not and clockwise is this. So 45 degree to this, 45 degree to this, that's of magnitude six. We need to find the magnitude and direction of the force. We need to find the resultant of these to force foster. So let's make the components of these factors. Uh, so for the one with six newtons, that's going to be over here. That's going to be six. Sign 45. 6. Sign 45 degrees. And over here, that's magnitude is going to be, it's a company is going to be six falls 45 degrees. Likewise for the other dog, that's going to be eight costs 60. That's going to be eight cause 60 degree. And this is gonna be eight signs 60 degrees. All right. So if if you have to find the resultant factor, that's going to be the some of these factors. So, for the one with six newtons, that's going to be six signed 45. All right. Plus six cause 45 J plus for the eight newtons. That's going to be eight cause or let's say eight sign for eight signs 60. But this is not my This is minus side Plus eight costs 60°. That's going to be in jail. So, if we add the lifetime six sign 45 is one over root two. So six over route I minus eight times route 3/2. Because Science sixties route 3/2. So, this is the component towards the eye. And for the J that's going to be six over root two Plus the cost 60 is one over to state what was just four. That was how it looks like We need to find a magnitude and the direction. So The magnitude is gonna be the same as the magnitude of these factors. So this is going to be square root of square root of 6 -8. Route 3/2 over. Uh Not really. That's six over root two minus eight times. Route 3/2 whole square plus six over Root two plus four. Who's Let me grab my calculator here. That's going to be six over to minus eight. Route 3/2 square plus. That's gonna be six over root two plus for Square meter. Taking the square root of this. So that's coming as 8.67 Newton's, which is the net force or the net uh net for the magnitude of the force which the walker must supply. And then all we have to talk or talk about the direction, the direction is just gonna be opposite to this one. So for this one we need we find a direction by finding the value of transit and that's going to be the y coordinate of the X corners. There's going to be six Route six over Root two plus four or six over root two minus eight. Route three or two. So if we talk about the angle tita that's coming as six over root two plus four, it shouldn't be divided by uh uh that's gonna be 6/2 -2 this if you take the pan and worse of this That's coming in as a negative angle. So negative angle means negative 71 .95°. So what does angle means? is uh the resultant as as coming either in the negative angle or if we talk about if you talk about the net magnitude net angle, that's that will be since the Y component is here and X component is here. So that's going to be somewhat like this. So the angle which data is showing is this one which is a negative side but the net net angle and the positive angle, it's like this. So the net angle towards the positive side, we're going to add 1 80 over here. So that's going to come out as 108.5 degrees. So this is the net net force, the the angle which the net force the dogs produced. So the walker must apply the force in the exact opposites. And so this is where the walker must apply the force. So that angle is going to be 71.95°. So we're going to write that, it should apply the force at 71 71.95°. Uh south of east because this is east and this is south. This much angle south of east. Uh Must the walker apply the force. Thank you


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