5

For mass m attached to spring with spring constant k, Newton's Second Law says the displacement of the mass, x (t), should obey the differential equation d2x k...

Question

For mass m attached to spring with spring constant k, Newton's Second Law says the displacement of the mass, x (t), should obey the differential equation d2x k = x(t) dt2 m If the mass crosses x = 0 at time t = 0, then one can show the displacement should be given by function of the form c(t) = A cos(bt) where A and b are constants and b is positive_(a) Compute the second derivative of x (t) = Acos (bt) using x (t) Acos (bt) . Do not include x (t) in your answer. d2(b) If the given differen

For mass m attached to spring with spring constant k, Newton's Second Law says the displacement of the mass, x (t), should obey the differential equation d2x k = x(t) dt2 m If the mass crosses x = 0 at time t = 0, then one can show the displacement should be given by function of the form c(t) = A cos(bt) where A and b are constants and b is positive_ (a) Compute the second derivative of x (t) = Acos (bt) using x (t) Acos (bt) . Do not include x (t) in your answer. d2 (b) If the given differential equation holds, then how does the constant b relate to the spring constant k and the mass m? b =



Answers

A mass $m=1.00 \mathrm{~kg}$ in a spring-mass system with $k=1.00 \mathrm{~N} / \mathrm{m}$ is observed to be moving to the right, past its equilibrium position with a speed of $1.00 \mathrm{~m} / \mathrm{s}$ at time $t=0$ a) Ignoring all damping, determine the equation of motion. b) Suppose the initial conditions are such that at time $t=0,$ the mass is at $x=0.500 \mathrm{~m}$ and moving to the right with a speed of $1.00 \mathrm{~m} / \mathrm{s}$. Determine the new equation of motion. Assume the same spring constant and mass.

For the following problem, we want to show that by using the chain role that X. Ft satisfies the differential equation. So what we are going to do is find the second derivative of um acts with respect to T. So the first derivative, if we have X. Not person omega T. Which were called W. T. Plus V. Not over W times the sine of omega T. And what we see is that by taking the derivative will have ex not um negative X. Not and omega T times omega I was using chain role and then we'll have plus Vienna over omega. Um Since omega right here, this is actually gonna be product rule. So we're gonna have the derivative of this times this plus the derivative of this times this. And we're going to want to use the chain the chain role within that to find the first derivative.

So we have the angular frequency of the oscillator. Omega equaling the square root of the spring constant, divided by the mass. We know that the general function for the position as a function of time would be equaling to the amplitude, multiplied by sign of the angular frequency multiplied by time plus the phase shift data. We can say at time equaling zero. We know that. Then Ex at T equaling zero seconds Would be equaling two. A sign of omega multiplied by zero plus data, sign of zero. It's gonna be Rather Tha Tha multiplied by zero is going to be zero. And so we have than that the sign of data is equaling zero or data equals zero. And so the function of velocity. So velocity as a function of time would be going to omega a. This would be co sign of the omega T. Plus the phase shift data. We know the phase shift is zero. So we can simply say that The velocity at equaling zero seconds would be then equal to omega a omega times T. T equaling zero would be zero. The phase shift is also zero coastline of zero. It's going to be then 11 times omega times a omega. So given that we can say that then A would be willing to be over omega at time equals zero. So this would be 1.0 m per second Divided by 1.00 radiance per second. And this is giving us of course one zero m. So the general equation for part A We have the X. as a function of time. This would be willing to 1.00 m. This would be multiplied by sign of 1.00 radiance per second, multiplied by T. We can say that then for part B At time equaling zero the displacement of the mass. We can say .500 m is going to be equal to a sign of data. So here we can say that the equaling omega a rather co sign co sign of omega cheap plus data. And we have done that. 1.00 m/s would be equal 2.00 radiance per second, multiplied by a co sign of Ceta. We can divide uh these two equations and we're essentially saying that then tangent Of data equaling .500. And so tha Would be willing to arc 10 of 5.500 26.6° or ingredients. We can save .464 radiance and so .500 m equaling a sign of data a would then be equal 2.500 m divided by sine of 26.6°.. And we find that the amplitude is then equal into 1.12 m. So for part B, the equation of motion is going to be equaling to 1.12 m multiplied by sign, and then this would be 1.00 radiance per second, multiplied by T Plus .464 radiance. So that would be your answer for part B. That is the end of the solution. Thank you for watching.

So we're interested in finding about a massive spring a little bit different from before, right? And we're told that it has a sort of infinite Tess Imo Mass that's given by here but that each part of it moves sort of in phase right with whatever the total v of the masses. Right? And so if I'm interested in finding the kinetic energy of something, we typically have just 1/2 MV squared. But the EMS and the V's are veering and depend on different things. And we have the EMS here, so I'm gonna actually do an integral here. And I want to integrate each individual infinite testable mass get into individual kinetic energy and that and them all up. But we actually just get to plug this in, right? So this is actually really not that bad. So we get em over l times DX. So that's the d M part. Let me have X V over l squared. And now we're integrating over DX. Someone integrate along with length of the spring. So I'm gonna pull out the appropriate Constance here. What? You're gonna see you get a 1/2 MV squared and then randomly, maybe not surrounded my l cubed in the bottom. Then we're gonna integrate from zero l. And the thing that's left over is X squared and e X, But this is 1/3 l cubed eso that l cube and that cube cancel into the Connecticut or give just the spring is actually gonna be 1/6 MV squared. And then we do have the mass still oscillating at the end of it. And that's 1/2 and B squared still right. So this is the new term were also asked to identify what happens to the period. I'm gonna kind of cheat here. All right, so here's our to buy em over. Okay, I want you to notice that the spring has a kinetic energy of 16 mv squared, right? It's sort of what if I suggestively wrote it like 1/2 AM over three b squared? It behaves as if it has a mass that's actually like 1/3 of it, right? And that's because parts of it are moving slower in parts of removing faster. But I'm going to use that over here. All right, so we need the total mass. Normally, we would just add them together, right? So I take the block. But I'm just gonna treat the spring as if it only had 1/3 of its mass, because that's what it looks like over here with the kinetic energy. When you actually do it, this actually ends up being the right answer.

Different. This is the problem based on our solution off. Lower discipline. Here it is. Given a block off mass M connected by spring. The mask off block is capital. Them mass anti sprig this morning, Kate, Is that spring constant? L is equilibrium. Debt? Yeah. All parts of the spring are vibrating in faith. The velocity off segmented TX yes, ex upon l. And to be well, a city off segment dx boom possible. Thank you. Here. Mass off segment is Do you have medical toe? Find kind of technology of the system? E back better. Um, I never see you, girl. Okay. Uh huh. For each segment, Can you take energy between Hold on? Uh huh. Uh huh. You know, deep K E hop, D m. And to be existed substitutes of it. Uh huh. Yeah. Mhm. Okay, d m is given, um, upon L d X and where the city is given E squid so geeky you will get hall M v square access square D X to pay dr Kind of dignity off block a spline system. Who kind of technological block, half and be script. Uh huh. Uh, mhm. Let me correct here and this could be upon L Cube for the limit. Do you don't do any integrating and solving it Total kind of technology off the system. We will get one by two Mm bliss Come by three be square. This is the answer apart. A no answer up be part Omega is called k Apart. Effective mass that is take upon M plus M by three. So time period off oscillation will be to pipe Blew it off M pless M by three upon cape That's all. Thanks for watching it.


Similar Solved Questions

5 answers
22_f(x, Y) dA = ? given f(x, Y) = cos(2x + y) and region is bounded by } < x < f- 1<y < 2x.a) ~ ~416~.693~.9485.1 e) None of these
22_ f(x, Y) dA = ? given f(x, Y) = cos(2x + y) and region is bounded by } < x < f- 1<y < 2x. a) ~ ~416 ~.693 ~.948 5.1 e) None of these...
5 answers
9 = /s)4 (h-A)_Y 27 xp =()h ' o=78- [z+ TF 9+28 = (1) h R*x8 = T 31192J Jn) ~^ 1- +I 'L7c| oS
9 = /s)4 (h-A)_Y 27 xp =()h ' o=78- [z+ TF 9+28 = (1) h R*x8 = T 3 1192J Jn) ~^ 1- +I 'L 7c| oS...
5 answers
Moving another question WIl [email protected] this responseQuestion 12SCF E3 ubiquitin Egase: mavatacn Ubiouitins some hormione receptors incuct(neir dcerudation may Induce cegradation 0t AUXIAA proteins allo Cerepresston FARF IYFC [ronscriptlon factors may induce degradation of DELLA-type repressors activating GA signallirg D,all oftne aboveVoving t0 another question will gJve Uis responsele1n8bd
Moving another question WIl [email protected] this response Question 12 SCF E3 ubiquitin Egase: mavatacn Ubiouitins some hormione receptors incuct(neir dcerudation may Induce cegradation 0t AUXIAA proteins allo Cerepresston FARF IYFC [ronscriptlon factors may induce degradation of DELLA-type repressors activatin...
5 answers
8. A charged particle moving in the plane of the paper enters a region with constant magnetic field that is perpendicular to the plane of the paper; as shown_ What is the charge of the particle and what is its final speed compared to its initial speed?X XPositive_ slowerB. Positive, the same C. Positive , faster D. Negative, slower E. Negative, the sameX X B X X X X
8. A charged particle moving in the plane of the paper enters a region with constant magnetic field that is perpendicular to the plane of the paper; as shown_ What is the charge of the particle and what is its final speed compared to its initial speed? X X Positive_ slower B. Positive, the same C. P...
5 answers
0 Provide your ~ne erminal 0 cos 0 dnswer side below: 'angle 0 intersects the unit circle the first quadrani dtaWhat are the exact
0 Provide your ~ne erminal 0 cos 0 dnswer side below: 'angle 0 intersects the unit circle the first quadrani dta What are the exact...
5 answers
Une iem7>n 2For the following uniform density curve, what is the probability that the random variable has a value of exactly 6?Jout ofquestion1250.725Correct answer is not listed1.00.125
une iem7> n 2 For the following uniform density curve, what is the probability that the random variable has a value of exactly 6? Jout of question 125 0.725 Correct answer is not listed 1.0 0.125...
5 answers
Thehac Simoson DameoinaMainjftrktt = #Uxo) + ro + 26)] 00"(c)Use the Composite Simson Parabola Rule with points = aslimalesin( Jx JdrSimpson'$ Rule - estimate the How Many points (roughly) would have be used 10-42 assume that fl+) sin( J) above Win . an elou (Ignore round-off error integra dervatives bounded by 20)
Thehac Simoson DameoinaMain jftrktt = #Uxo) + ro + 26)] 00"(c) Use the Composite Simson Parabola Rule with points = aslimale sin( Jx Jdr Simpson'$ Rule - estimate the How Many points (roughly) would have be used 10-42 assume that fl+) sin( J) above Win . an elou (Ignore round-off error in...
5 answers
Fo CucooneCompute the gradient of the function at the given point f(x, Y) = 8x2 _ 9y; (-5,2)0 A: 200i = 18j 80i ~ 18j 400i ~ 18j 0 D: 80i ~ 9jClick to selec your answer;
Fo Cucoone Compute the gradient of the function at the given point f(x, Y) = 8x2 _ 9y; (-5,2) 0 A: 200i = 18j 80i ~ 18j 400i ~ 18j 0 D: 80i ~ 9j Click to selec your answer;...
5 answers
MaliX Zevo matnces C 2 X2 non- Xamlle buf B#c an AB = 0 4B = AC cive such thck A , B , and C
MaliX Zevo matnces C 2 X2 non- Xamlle buf B#c an AB = 0 4B = AC cive such thck A , B , and C...
5 answers
Consider the following supply and dernand system with goodsQ4l+MP 4 Pz biP bPz CP CPz diPy dPzQ4What are the equilibrium conditions for this system? Hint: there are of them | Let'$ assume that goods and 2 are substitutes, like Coke and Pepsi_ What sign would we expect the following terms t0 have (positive Or negative)?
Consider the following supply and dernand system with goods Q4l +MP 4 Pz biP bPz CP CPz diPy dPz Q4 What are the equilibrium conditions for this system? Hint: there are of them | Let'$ assume that goods and 2 are substitutes, like Coke and Pepsi_ What sign would we expect the following terms t0...
4 answers
Inveree Laplaco Ibe8+function F (8) 76-16-2) transform of the 'find the inverse Laplace Problem 1: differential equation solve the
Inveree Laplaco Ibe 8+ function F (8) 76-16-2) transform of the 'find the inverse Laplace Problem 1: differential equation solve the...
5 answers
Let X be a random variable that can take the values 5, 8,or 13.And let f(x) be its probability distribution:Find the variance of the random variable X, given that its probability distribution is f(5) = 0.1f(8) = 0.4f(13) = 0.5Round your answer to an integer:
Let X be a random variable that can take the values 5, 8,or 13. And let f(x) be its probability distribution: Find the variance of the random variable X, given that its probability distribution is f(5) = 0.1 f(8) = 0.4 f(13) = 0.5 Round your answer to an integer:...
5 answers
N3" converges or diverges (n + 1)!point) Use the ratio test to determine whetherFind the ratio of successive terms. Write your answer as fully simplified fraction. For n > 9,4u+1 lim 0 'COlim n+CC(b) Evaluate the limit in the previous part. Enter & as infinity and 0 as -infinity If the limit does not exist, enter DNE an+] lim Is an(C) By the ratio test; does the series converge, diverge , or is the test inconclusive? Choose
n3" converges or diverges (n + 1)! point) Use the ratio test to determine whether Find the ratio of successive terms. Write your answer as fully simplified fraction. For n > 9, 4u+1 lim 0 'CO lim n+CC (b) Evaluate the limit in the previous part. Enter & as infinity and 0 as -infinit...
5 answers
Piciuie eheGA_0 pT 20 pT 40 pT 60 pT 180 pTIRcn19. What is the direction of the magnetic field at the black dot shown in the picture? l=h=6A, To the right To the left Into the page Out of the page There is no direction because the field is 0Jcm71zQuestions 18 und 19
piciuie eheGA_ 0 pT 20 pT 40 pT 60 pT 180 pT I Rcn 19. What is the direction of the magnetic field at the black dot shown in the picture? l=h=6A, To the right To the left Into the page Out of the page There is no direction because the field is 0 Jcm 71z Questions 18 und 19...
5 answers
Series Me Determine "-0 (n+1)3" te thei interval of convergence and radius of convergence for the
series Me Determine "-0 (n+1)3" te thei interval of convergence and radius of convergence for the...
1 answers
A woman weighs 120 lb. Determine (a) her weight in newtons $(\mathrm{N})$ and $(\mathrm{b})$ her mass in kilograms $(\mathrm{kg}) .$
A woman weighs 120 lb. Determine (a) her weight in newtons $(\mathrm{N})$ and $(\mathrm{b})$ her mass in kilograms $(\mathrm{kg}) .$...
5 answers
STUDY GUIDE MOLESSIUDY GUIDEC For the reaction ofKI25 + 2 KzS, how many AtOms of sulfur will renct with 52 atomsFor the reaction AAl 302 7 2Al,O how many moles of oxygen will react with 41 moles of aluminum?
STUDY GUIDE MOLES SIUDY GUIDEC For the reaction ofKI 25 + 2 KzS, how many AtOms of sulfur will renct with 52 atoms For the reaction AAl 302 7 2Al,O how many moles of oxygen will react with 41 moles of aluminum?...

-- 0.023329--