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Problem bung from (20 tlie pts } f 2 N #prig torce of when the Md Its Motiol suetchies Lst Iiu 4 equilibrium Ve-locity damped SpJng ([ position and then m/s_ Visc o...

Question

Problem bung from (20 tlie pts } f 2 N #prig torce of when the Md Its Motiol suetchies Lst Iiu 4 equilibrium Ve-locity damped SpJng ([ position and then m/s_ Visc oe TS Telast| Tuit ially, Ile Mass dnuj" tliat Use the from rest "Xerts pushed fotee Xutoli Wp' 0.5 When 3"Mllw % ahovo there O externa] dlerive #I foree. Hiut ; "quation 6) Find 'luu the Ku(t)s Meaheels the displaccment It jO ofl 30"(4), 1myh "(t) of tle Axsuing tlie Mass at AHY time (reqquency

Problem bung from (20 tlie pts } f 2 N #prig torce of when the Md Its Motiol suetchies Lst Iiu 4 equilibrium Ve-locity damped SpJng ([ position and then m/s_ Visc oe TS Telast| Tuit ially, Ile Mass dnuj" tliat Use the from rest "Xerts pushed fotee Xutoli Wp' 0.5 When 3"Mllw % ahovo there O externa] dlerive #I foree. Hiut ; "quation 6) Find 'luu the Ku(t)s Meaheels the displaccment It jO ofl 30"(4), 1myh "(t) of tle Axsuing tlie Mass at AHY time (reqquency Viscos dumpet cd of the svstem? is removed from What the xystvm (7 = external 0), what is the natural exhibiting force should be resonant pehavior ? applied to the SvStem Mle interestexdl tlie systen



Answers

A particle of mass $m$ is attached between two identical springs on a horizontal frictionless tabletop. The springs have spring constant $k$, and each is initially unstressed. (a) If the mass is pulled a distance $x$ along a direction perpendicular to the initial configuration of the springs, as in Figure $\mathrm{P} 8.47$, show that the potential energy of the system is $$ U(x)=k x^{2}+2 k L\left(L-\sqrt{x^{2}+L^{2}}\right) $$ (Hint: See Problem 66 in Chapter 7.) (b) Make a plot of $U(x)$ versus $x$ and identify all equilibrium points. Assume that $L=1.20 \mathrm{~m}$ and $k=40.0 \mathrm{~N} / \mathrm{m} .(\mathrm{c})$ If the mass is pulled $0.500 \mathrm{~m}$ to the right and then released, what is its speed when it reaches the equilibrium point $x=0 ?$

If this distance is X this is L then that distances elsewhere to us X square on things distances Tetteh So we need to know what is the highest under the comfort of the force ethics. So FX equals if who signed Tetteh on f is que down to the extension and go Santa is eso You will come out to the negative zero to x There are two off the springs so two f x t x equals negative toe integration zero to eggs K x minus que el X over square root off elsewhere plus X square in traditional p x and this will come out to be gay X square That's what the blast site now K X squared plus to K l l minus experts us l squared. Yeah, putting the values in, we would have Yeah, the U versus X. We look like this on DFO or the last part. We would have initial kinetic energy plus initial potential energy equals have times mass 1.18 kg times the square we final square So we finally will come out with 0.8 to 3 meters per second

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

To solve this problem. I draw the diagram but just to get it carefully this angle is alpha. Here I can write devaluate empty and in this side the normal force and it's working and here it is what is working. So I can write the value of fx is equal to M. Omega psI X. The value of N. C. Jed is equal to I C B Jed. And I can also write mg sine I'll fight the minus apple is equal to M. Omega rated. Big question number two and late Let it be a question # one. Not to let it be question number one and F. R. Capital R. Is equal to two by five. M. R. D squared vita This education number two going forward I can write the value of fr is smaller than equal to KMg glass alpha. In addition the absence of sleeping provides the kind medical relationship between the acceleration. Omega is equal to beat our this is the question # four. So simultaneously solution for the four equation. All the four equation gives to care costs Alpha is greater than equal to two by seven saying alfa Or I can write case greater than equal to two x 7 than alpha. Now solving part B. So using the equation one and two of the party we can write omega psi is equal to fight by seven G. Sine alpha. Also the value of V city is equal to Omega City which is equal to five x 7 G. Shine. Al fatty lated vacation number. Fight. Hence the south kinetic and disease given by T. Is equal to half M b C squared plus one by +22 by five um rd squared omega squared. We turn for the simplification, I can write the value age seven x 10 M. B c squared, Which is equal to seven x 10 AM. Multiplication, five x 7 G sign al Fatty Police Square, which is required to fight by 14 MG Square, Shiny Square al 40 square as the answer.

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.


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