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Previous Problem Next ProblemProblem Listpoint) weight is suspended from the ceiling spring: Let d be the distance in centimeters from the ceiling to the weight: Wh...

Question

Previous Problem Next ProblemProblem Listpoint) weight is suspended from the ceiling spring: Let d be the distance in centimeters from the ceiling to the weight: When the weight is motionless, d = 5 cm the weight is disturbed, it begins to bob up and down, or oscillate _ Then d is periodic function the time in seconds_ so d = f(t)_ Consider the graph of d f(t) below; which represents the distance of the weight from the ceiling at timeCocaIaet)Based on the graph of d = f(t) above which of the sta

Previous Problem Next Problem Problem List point) weight is suspended from the ceiling spring: Let d be the distance in centimeters from the ceiling to the weight: When the weight is motionless, d = 5 cm the weight is disturbed, it begins to bob up and down, or oscillate _ Then d is periodic function the time in seconds_ so d = f(t)_ Consider the graph of d f(t) below; which represents the distance of the weight from the ceiling at time Coca Iaet) Based on the graph of d = f(t) above which of the statements below correctly describes the motion of the weight as it bobs up and down?



Answers

An object weighing $W$ pounds is suspended from a ceiling by a steel spring. The weight is pulled downward (positive direction) from its equilibrium position and released (see figure). The resulting motion of the weight is described by the function $y=\frac{1}{2} e^{-t / 4} \cos 4 t,$ where $y$ is the distance in feet and $t$ is the time in seconds $(t>0)$. (a) Use a graphing utility to graph the function. (b) Describe the behavior of the displacement function for increasing values of time $t$.

All right, let's use the graphing utility for this problem. So here's my graphing calculator and I'm going to check the mode and make sure I'm in radiance and then we go into why equals and this is where you would type in your function. It was wyffels 1/2 e to the negative t over four times co signer fourty so the calculator might use an X instead of a T. And then let's set a good window. So we're told that T is greater than zero. So I'm going to go with 0 to 10 on my X axis, and then I'm just gonna leave my wife. Values is negative. 44 we'll see what happens. And if we don't like it, we can change those. So here we see that graph. You see the oscillation as the weight is on the spring going up and down, and we see that over time there's less than last displacement. So for Part B, we would say that the displacement is heading toward zero as time increases

Okay, so we haven't object on this spring, and this object is 10 grams, and we pull it down five centimeters, and I have the damping factor. The spring has a damping factor of 0.8 grams per second and with simple harmonic motion. The period of this is three seconds. And we want to find the function that represents the motion of this object. Well, because we have a damping factor. And because I have written here because we have a damping factor, we know that we're gonna be using damped motion, okay? And the formula for damp motion is given right here. Um, so all we need to do is determine the values we need to plug into this formula. We A b in in omega. So let's go through. Figure all this out. A is defined as the starting position of our object. With respect, Teoh, it's resting position because we pulled it down five centimeters and note that we pull it down a will be negatives. Five. When it took negative because they pull it down. Okay, next B B is defined Is our damping factor that we have that given right here. So be is zero cranked. Eight Okay, on the next one is m Okay. Our next one is M m. Is defined to be our mass, so m will be 10. Okay, All that's left is Omega, and we're not given omega strictly over here. However, we do know that one on the way. The damp motion, it simple harmonic motion are defined. Omega relates to the period in that period is two pi over omega. And because our periods three this equals three so we can use this equation saw for Omega. Okay, Our first step will be to multiply both sides by omega. That will give this to pie equals three omega. And then the soft Romania we could by both sides by three. So that gives us too high over three equals omega. Okay, so now that we have all four of our values that we need, we can go ahead and play it into art form. So let's do that. We have dft equals a, which is negative five times e to the negative B, which is 0.8 times t divided by two times down. M is 10 so two times 10 is 20. Okay, Now, Times Co sign Oh, square rates of Omega Squared Omega is two pi over three. So two pi over three squared is four. I squared over. Not ok. Minus B squared V is 0.8, So b squared will be zero point 6/4. Okay, over four times in squared is 10th. So that means M squared is 100 times four is 400. Okay, All that times t. Okay. And that is our function. We need to graph it. But this is a very complicated function to grab my hands. So we're gonna use a graphic tool. I use Dez, Miss, and let's go over there and, well, she that this is what our function once like, direct up. Hey, started negative five. And we go up and down and we can see the damping factor. It's not as we go along, it's not going up is high. And that's not going down slow. Okay? And that is our function

Okay, so we have an object and you're gonna spray. And its massive 10 grams, we pull it down five centimeters. And the dampening fact, the damping factor of that spring is 0.7 grams per second. And the period, um, if we consider it in simple harmonic motion, the period of that objects motion is three seconds. We want to describe the whole notion of this object with a function. Well, since we have and you can kind of see the answers goes way over here. Since we have a damping factor of 0.7 grams, we know that we're gonna be using dance motion, and the formula for damped motion is given right here. So all we have to do this fine. The values that we need a B in an omega employed them into our formula, and that will give us the function that describes the motion for this object. Okay, so let's start with a a is the starting position, Um, with respect to the objects resting position. Okay, so since we pulled down, five centimeters are starting position. A will be negative five. Okay. And the next one we wanna find is B B is defined to be our damping factor. So we already have our damping factor right here. It 0.7. So be is he is 07 Okay, now the next one is I m is defined to be our mass our masses already given this 10 grams so m is okay. The only one left is Omega. Now Omega isn't strictly given over here, but we know with regards to damped motion and even simple harmonic motion Omega is defined with respect to the period. And we know that the period in terms of Omega is two pi over Omega. Okay. And since that's our period, we can set that equal 23 Now, using this equation, we can solve for Omega, the first step will be the multiple. Besides fine omega and I dont give us two pi equals trio omega and to solve for Omega. We divide by three on both sides and we'll get to pie over three equals omega. Okay, Now we have to do is plug these values into our formula. So let's do that. So we have DFC equals negative five looks okay. D of t equals negative five times e to the power of negatives. B, which is 0.7 times team over two times m m is 10. So too will be 20. Okay, And we must, by all that times co sign It's very, uh, Omega squared. Omega is two pi over three, so omega squared will be four pi squared over None go minus B squared via 0.7. So b squared 0.49 over four times m squared m is 10. So M squared is 100 times four is for okay and all that times t and that's our function right there. We want to graph it. However, this is a pretty complicated function, a graph. So we'll use a, uh, graphing tool to grab it. I've chosen business. And here I plugged it in and we can see five periods of our graph and you can see it's you can see the damping motioning. The damping motion happened. It's not going quite as high as it ever gets, and it's not going quite as lows it's gotten before. And it's staying between negative 55 Okay, and that is our phone

We're discussing harmonic motion. So the height obtained by a weight attached by spring set in motion is a of T is equal to negative for cosign of a pie t. So we need to find the maximum height that the weight rises above the equilibrium. If f sorry s f t is equal to zero. Okay, so the maximum height it can be is when co sign of a pity is equal to one. Because remember that cosine is between negative one and one, so the greatest it can be is one. So that means at the greatest height it could be S f T is equal to mhm. So negative four. But in this case, this is the lowest. So let's find remember. So if the lowest is negative for the highest is four great part B. When does the weight first reach a maximum height if t is greater than or equal to zero? So it reaches its maximum height when four is equal to negative for cosign of a pie t. So this tells us that co sign of a pie T is equal to negative one. So that means ate pie tease. You goto pie because co sign of pies Negative one means t is equal to pi Over eight pie just won over eight in seconds or C is what is there the frequency and the period So recall that the frequency is equal Thio, make over two pi Oh, Magda is equal to eight pie So eight pi over two pi is it go four. So the frequency is four cycles per second and then the period is one over the frequency so the period is 1/4 seconds.


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