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At least one of the answers above is NOT correct:point) A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimet...

Question

At least one of the answers above is NOT correct:point) A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time 0 , the resulting mass-spring system is disturbed from its rest state by the force F(t) 90 cos(8t) . The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds:Determine the spring constant k. 1000 Newtons meterFormulate the initial value problem for y(t) , where y(t)

At least one of the answers above is NOT correct: point) A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time 0 , the resulting mass-spring system is disturbed from its rest state by the force F(t) 90 cos(8t) . The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds: Determine the spring constant k. 1000 Newtons meter Formulate the initial value problem for y(t) , where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction: (Give your answer in terms of y, y' ,y",t.) Differential equation: -1OOy-9cos(8t) help (equations) Initial conditions: y(0) and y' (0) help_(numbers) Solve the initial value problem for y(t) y(t) 4(cos(8t)-cos(1Ot)) help (formulas) Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 <t < 0 . If there is no such maximum_ enter NONE maximum excursion NONE meters help (numbers)



Answers

Suppose an object of mass $m$ is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position $y=0$ when the mass hangs at rest. Suppose you push the mass to a position $y_{0}$ units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system , the position y of the mass after t seconds is
$$y=y_{0} \cos (t \sqrt{\frac{k}{m}})(4)$$
where $k>0$ is a constant measuring the stiffness of the spring (the larger the value of $k$, the stiffer the spring) and $y$ is positive in the upward direction.
A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by $y=10 e^{-t / 2} \cos \frac{\pi t}{8}$
a. Graph the displacement function.
b. Compute and graph the velocity of the mass, $v(t)=y^{\prime}(t)$
c. Verify that the velocity is zero when the mass reaches the high and low points of its oscillation.

Let's discuss the set up for this problem were given the position of the mass that suspended from a spring, it's going to be why equals why not? Okay at times the code sign of tea route K over em. So we know that K. Is going to be the stiffness of the spring. And we want to find Dy DT ross has to find other values. Um so we see that if we look for Dy DT, we're also going to want to eventually find um the second derivative as well. So what this is gonna look like is Dy DT will be um taking why not? Than the co sign that's going to change to a negative sign of this whole thing. And then we're gonna multiply that by the um derivative of this internal component. But we see that this internal component is just a constant right here with the variable right here. So we'd end up just multiplying it by this right here. And then we do the same thing when calculating the second derivative

Hi everyone at sundown and figure so late most of em it displaced to the position. Yeah, explain the large Erion of the system. Very big kind of technology minus potential energy. Half an extra 40 square glass half empty my daughter script because half keith by square plus help plus x squared minus and not less half Okay. By square plus l minus X. Foley square minus and not squared to find equation of original motion. Youth. The and your lower education. Mm It would be the upon entity still had over days. Extort called Today Neocortex. Okay. Mhm. This will give you I am acceptable dot it's called Mine escape. A root of wise calculus and plus X fully square minus and not into help plus X over and who taught by square plus and plus access by and this weekend right my escape the rule car by square plus. This is minus king by square plus and minus X. Holy square minus and not in two and minus X over by square plus and minus X squared minus. But yeah. So this could be realized Yeah. Mine escape 1- and not route of by square plus. Helpless sex. What is square and two helpless sex. Let's keep bond minus and not over. Vice squared pleasant and minus X. Holy squared into and minus X. And this result could be simplified to um my escape two x minus. I'm not into it bon appoint Rudolph Y square plus and plus access by minus. Burn upon a road off by square plus and minor sex. Holy square minus and not X. Burn upon by square plus And plus six square blessed one divided by route of by square plus and my next access square for hell plus minus X. To be greater than by This will be approximately two am acceptable. Dot is called to Cape two X- and noted. Born upon helpless X upon and minus X. And finally you can Right I am extroverted to be mm mhm might escape two x minus and noted I'm going over helpless six minus all over at minus x minus and not X. Born over help plastics upon one minus a leg up on X. And this could be approximated to minus two cakes. This is the equation along X axis. You will get M X two K X. S. Call to zero. This is 1st part, no second part. Do you hold our duty of they did this very over. They'll buy dot you didn't buy. Mm. Yeah it will give you an inevitable dot. It's cart Okay root of by square plus help plus X. Holy square minus and not and two why upon by square plus Helpless six. Holy Squaring. And this could be simply fight too. I am by double dot let's go. Ok one minus And not upon her into vice for 20. Yeah she put the ratio of yeah peters okay. Equal to T X upon T way. Okay. Oh America by upon America it's a rule of cape 1- and not by ed divided by m upon rule cough to give I am so it can be written as the root of 1- and not buy it deep part and the mostly start from just over the city. Uh huh From and Jiro velocity the position of mass it described by access code to be not sign up Omega XT and bicycle to a not because of make a white jeep. Mhm He bought by kicking off and exit grab our given its that's all. Thanks for watching it.

Okay, So question is a Kadima's attached to the lower end of the Spain stretched it 1.96 m a viscous medium over the resistance off four times that instant without 50 millimeters per second, the masses pulled to mutability Librium and Elizabeth. Velocity of 3 m per second downwards. Okay. And external force applied. 20 costume. Ellery Newton. We need to find the position of the mask at T equals toe by or at the end, off by seconds. Okay, so let's try the data for so two kg mass for Emmys to right now. This must lower the spring by 1.96 m, so m G force must be balanced by spring resisting force cakes. MH two G's 9.8. Okay, we don't know. And access 1.96 Okay, So if you multiply this bigot Yes. Two into 19 point eighties, 19 point 6/1 19.0.96 And that is 10. Okay. And now Delta, it is given its four times a discussed 10 ties for okay. And a 40 is already given directly. 20 cause t. Okay, so now we can form our equation. So we know it is. And why Cabrera's plus Delta y dash plus k y equals toe foo d m s too. So to fight Ebola spread Delta is four wide ash, the skin sistan wide and this is 20 cause Great. So absurd equation will be to r squared plus four r plus 10 days 20 Costea will have to solution the complementary and particular solution. So, for complementary, we replace the right inside or external force with zero. This is quality equation solving for our gives us minus one plus minus two s. This is Alpha. This is Omega. So we get our Y C s. It is two minus tea party. All right, See you one first. Duty. Bless you. Too scientific. Okay, Now for particular solution, wipe. This must be in the form of cause and sign. Right? So a scientific plus B costly. So why Nash off? They will be a cost e minus B scientific. The transition, of course, is minus. And why the world I saw he will be minus a scientific pleasant, because thing now, this is the solution of the defense allocation. Therefore, it will satisfy this. Okay, so we can right here. Let me separate this part. So two times minus a society pleasant be Costea. Okay, plus four times a costea. Okay, we're subjecting this virus minus B society last 10 times. Why is this so? A 70 present in costly is the question to cost a great. Okay, so now the club sign and horse taking scientific common from all systems. So here we have minus two way from here we have minus four b and for me over yet. Danny, Similarly taking Costea common from here. We have to be from here we have plus 40. And from here we have and And this is equal to two costea plus zero scientific, since we don't have the same time, So I'm comparing this must be caused to zero. So to a plus 10 3 minus to a plus 10 is a minus four b cost a zero and to be percentage will be plus for it equals toe to on solving this. We get a easy question and I'm sorry we made a mistaken Mrs minus, so we'll distantly minus. So the big cost, um, will be minus b. Okay, so we'll have your minus to B plus 10. 80 okay. And this is equals to two. So not too. Because this is 20 Costea, right? Our function is 20 costea. Okay, so after doing the necessary genius So we have the Selectmen righted fish eight B plus four A equals to 20. Now it's all in the situation. We get easy cause to one and he was toe So therefore her y p is society yes to cost. Okay, so therefore the total solution will be Why questo? Why simples y p and why is he was the one Teddy those It is two minus t c one because duty plus C two sign duty and wipe a scientist plus two Caustic right? This is why off t not differentiating this with respect. Oh, we give, we get it is to minus t differentiation of this will be minus even signed to tee times too plus C to cause duty. That is all these times too. Plus see one caused duty plus e to sign duty time Definition of this is it is to minus 10 times minus one Plus the transition of sign is costing and the transition of two courses minus to sign now applying the boundary condition or initial condition. So why off? Zero is why zero is, too. And why dash off the history case? We'll apply this one to this. And let's do this one. So let's do that. So to ease equals two. It is two minus zero. See? One cause zero. Let's see to sign zero plus sign zero plus two times because it Okay, so what? We get here to ease see one and see too easy. Right? Since this is zero, here's Edo and this is to so to get canceled, we gets evenness. Is you now to check for second one will apply these conditions. Three Ys equals two. Here it is. Two minus zero minus e. When sign zero times two plus C. Two times called CEO times two. Let's see one Constancio lesson to sign zero. And this is it is to minus 10 to minus one. Let's cost zero my next two times Sign zero. So this gives us think this is one police say zero this time is zero. And this time it right. So here we have left with to see two. So this is one. Here we live with C one times minus one minus C one and he developed a plus one. Okay, but see when is zero? So we have three is equals to two C two plus one. So too is to see two. So therefore C 321 Okay, so we got the value off to now. Our final solution is why Off t equals to it is to dynasty signed duty plus scientific plus to cause D Now with no value off while fire. Right. So why a five will be here is to minus five. Signed to buy. Let's sign right. Plus two times close by. Signed by zero signed by zero. So they're left to cross benches minus two. So that is 2 m above the supreme, right? So why By minus two? So this is to Mita a boat equilibrium. Thank you.

So in this problem, we have a diagram, and I'm gonna reproduce the diagram in pretty good detail because I think this problem is going to be kind of confusing. This is the X axis now on the Y axis. This is where is attached up at the top of the bottom. But it starts here, the particle and its unstrapped. And then it comes out to here. And we're saying that this distance that it is stretched is X No, it's unstrapped distance for the spring was l l. So when I draw the spring this way and the second spring this way, then the first thing I want to dio is I want to find, um this distance. Um, I'm gonna call this l two, so using the Pythagorean Theriault L two is the square root of l squared plus x squared. We also know that force equals okay, X, but this force is going to be in this direction or this direction. And I actually want to write del that X so we don't get confused between exes and whatever in here, this is just the generic formula. So the change in the length So in this problem. The force is k times the change of the length, which would be l two minus l. But L two is the square root of l squared plus X squared minus l. Now there is a component of each of these forces that is in the negative X direction. The force in the X direction equals that that force times the co sign of fate. But we know that the co sign of Fada is just the adjacent which is X over the high pot news, which is l two adjacent overhype oddness. And we also know that L two is the square root of l squared this X squared now in the X direction. This force is negative. So the force is in the negative direction and there are two of them equal to K. So I'm just putting this into here and multiplying by negative too. So that's going to be I still have the axe. Then I have the square root of l squared plus x squared minus l over square root of l squared minus X square. All right, negative two k x. I'm just going to distribute this to the first term and the second term, one minus L over the square root of l squared minus X squared. And that is in the X direction. So that is our answer. And now there's a part B You got to go to the next page to see Part B. We need to find the potential energy. All right, well, the potential energy is the opposite of the integral off the force D x. So the potential energy is I'm going to distribute the two K x again the integral of to Okay, X minus to K L X over the square root of l squared minus X squared DX. All right, So, um, let's see if we can do this. You of X is well, the integral of axe D X is X squared over two, so that first term becomes okay, X squared. But the second term is minus to okay, oh, times the integral of X over the square root of l squared minus x squared D x. I just want to make sure that I didn't make any mistakes already. Um, negative two k x. And it looks like I did No, I didn't que x two k x and minus No, I didn't make any mistakes yet. Um, minus two k l the integral of X over l squared minus X squared DX. All right, so now the difficulty is what is the integral of X Over. Whoops. The square root of L squared minus X squared DX. Well, I'm going to start by letting eggs equal l tangent of fada. Therefore, D x would be l driven. The tangent is the sequence squared of Fada di Fada. So if I substitute that in, I get l tangent of Fada over the square root of l squared minus l squared Tangent squared of Fada and I replaced d X with l seek and squared of Fada di Fada. All right, so that's gonna be the integral, Um, notice that the numerator, while numerator still just l tangent of Fada. But I have two l's there, so I'm gonna write l squared and then I've got seeking and squared of Fada de theater. But the denominator is now when a factor out l squared. So it's gonna be the square root of l squared times one minus tangent squared of Fada. All right, square root of l squared is just l So that's going to leave me with an l only one l Tangent of fade Seek and squared of fade Uh, over the square root of one minus the tangent squared of state. Well, that would be the square root. One minus two tangent squared is the sequence squared. Di Fada, which is gonna give me l times This integral seeking squared near the square root of the Seacon Square is just the seek. And so this is just going to be the tangent of fada times this seek it of Fada d theta. But now l tangent is sign over Cosa seeking to his one over co sign. So this is the same as L into Girl of the sign of Fada over the coast sine squared of data de Fade. So now I'm going to let you equal the co sign of data. So do you. Driven of the co sign is the opposite of the sign de theater. So now it becomes l in to grow in the numerator. I have sign of data d theta. You can see it right here. Sign of data death data. That's the opposite of de you so I can write books opposite of D. U. And then in the denominator co sign squared of data is use squared. So that's going to be negative. L times the integral of you to the negative to power do you, which is negative. L times you to the negative One power over negative one. We think about this again. This is negative. L you negative one power of her negative one. Okay, good negatives. Cancel out and it's just l over you, which is the same as l over you was the co sign of data way up above. We said X equals l tangent Fada. If I draw a rate trying with this being tangent I mean, with this being Fada tangent is opposite over adjacent using the Pythagorean theorem. This is the square root of l squared plus x squared. So the co sign of Fada would be the adjacent over the high pot. Innis who sign is adjacent overhype Otterness. So this now simplifies Teoh l over oh over square root of l squared plus x square. The coastline of Fada is the adjacent overhype Otterness l over the square root of l squared, plus x squared. I was just leaves me with word of l squared plus x square. So I went through all of that, and I got the square root of l squared plus x squared. So let's keep going here equals Okay. X squared minus to K l. And then the integral of X over the square root of elsewhere in minus X squared D X turned out to be squared of l squared plus x squared. Okay, I am somehow missing a term. All right. I think that what I'm missing is that when I take this into girl, um, of way down here that I have to add a constant because this is not a definite integral. So that's gonna be l over you. Plus a constant, which gives me the square root of l squared plus x squared, plus a constant. And so up here we took this in a girl regardless, there's gonna be a constant having trouble writing it in there. Okay, so let's just keep going. I want to say, um, you as a function of X is que x squared minus to K. L times, the square root of l squared plus x squared, plus a constant and according to what's written in the book. That constant must be two K l Square to K l squared. Now we do know the one boundary condition that the potential energy at zero would be zero. So that's gonna be K zero squared minus two K. What? What whoops. Que el square root of l squared plus zero squared plus c zero zero this negative two k backwards l squared plus c soc would be negative two k l squared. However, the square root of L square that we see right here could either be pot plus or minus l. And so it is possible that this could be, ah, positive. And so that could give us a positive, which is exactly what we want it Not exactly sure why we need to choose the negative here. Um, I believe we need to choose the negative so that the potential energy would generally be positive. So nevertheless, um, we do come up with the answer using the boundary condition that you as a function of X, is que x squared. Plus two que el times l minus the square root of X squared plus l squared. And I do believe that this were choosing the negative here so that this ends up with positive numbers. Okay, let's move on to see, we need a plot of you of X versus X, letting l equal 1.20 que equal 40. And there's even a part d after this. Well, first of all, I'm just going to go to a graphing calculator and I'm gonna put this in. So, um, 40 times X squared, plus two times 40 times l, which is 1.2 times 40. No, El is 1.2. AL is 1.2 minus square it of X squared plus l, which is 1.2 squared. And I am kind of interested. What if we had chose the negative up here, and then that would give us a negative? L yeah, that's going to give us potential energy that's on both sides of the That would be potentially negative. So that is the reason why that one has chosen. Okay, so I'm plotting potential energy vs X. I want to look at the question again, and so X is going to be positive where there's nothing stated about moving X in the negative direction. So we read it again, Maybe there is. And I think you can assume that it's it's gonna be pot pulled in the positive direction. Looking at my graph one. I guess that would be meter race. This is gonna be the potential energy in Jewell. Son is gonna erase this. I'm gonna write jewels here. I'm gonna try to So five, 43210 Okay, one comma. Five is there? Zoom in a little bit at 1/2. Its basically at point four. Do you not see if I noticed anything? Nope. Okay, so let's work on the rest of this question. I'm still recording. Yep. Looking for the question. Equilibrium points. Okay. All right. So stable equilibrium is when the potential energy is a minimal. So right there is stable equilibrium. Unstable equilibrium is where the potential energy is a maximum, but it never reaches a maximum. Um, goes off to infinity. Neutral equilibrium is when the potential energy is constant. And that does not occur either. All right, last question. So that's right. This is D. So we're saying that X now is 0.500 meters and I need to figure out the speed at X equals Zeer. All right, so all of the potential energy is going to be converted to kinetic energy. So potential energy of the spring is going to turn into kinetic energy potential energy this spring. I just need to put in X equals zero. No, no, no. I need to put in X equals 0.5. So let me go to my calculator. X is gonna be 0.5 five. That gives me a value of 0.4. So the potential energy in the spring is zero 0.4 jewels that's gonna be converted to kinetic energy, which is 1/2 M. What were we using for em? I don't seem to have written that down, so let's look in the question. Ah ha is a very beginning. M equals 1.18 kilograms. Connect energy is 1/2 times M V squared, so V is going to be the square root of to time zero point for Jules over 1.18 kilograms. Back to the calculator, where route two times 0.4. Divided by 1.18 eight to three meters per second. Um, now it would be going in the negative direction. So negative 0.82


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