Mit Professor Walter Lewi's Physics 801 Lecture 30

you did have some problems with the physical pendants and i will do talk a little bit more about physical presence but first look at the picture in very general terms i have here assaulted object which is located about point b about an axis critical to the blackboard and here at sea is the center of mass you don't recognize m and sort of issue a force angie and that the separation being here i'm going to offset it over angle fade out and i'm going to also lady clearly that has to be a force of two p_m_ if there were no force that the big his object would be accelerated down which acceleration g and that's not what's going to happen but i don't care about that force because i'm going to take the port about point p remember when we had last spring he's the one i mentioned a case we have equal to an eight and that sort of spring became minus k action to minus sign indicates that it to restoring force so we don't get something very similar in rotation port becomes to work mastercard moment of inertia acceleration becomes angular acceleration so now we have minus our culture and the minus sign indicates that it is the story if i take it to work quantity of two point p that i have this is the position vector which has magnitude b reporters and he and have to multiply by this time of later so it'd be crimes and g i'm to sign a fade out and that now equals might decide to bring to mind is here minus the moment of inertia about point b binds all five and all feisty angular acceleration which if they don't double dot i bring them together and i'll use the small angle approximation small angles then decide a fade out is approximately fade out if data is in radians and so i bring his old one site so we get fade up double dot plus angie provided by the moment of inertia about that point pete pine fatalities by small an approximation equals zero and this is a well-known recreation it is clearly simple harmonic oscillation beta because this is a constant and so we're going to get every solution that fade out equal fayda maximum you can call that the angular amplitudes french coastline only got three plus fly there's only a goner is the angular frequency is a constant omigod here fade adults beginning of the velocity at which is not a constant the two are completely different that is the angular frequency so we know that the solution to this differential equation gives me all made up disgruntled all this constant but it is be and jean divided by the moment of inertia about that point p and sort of period of also mention two-parter fighter bar omega sort of his compartment i'd be scrambled of all right rather quick to point b provided by the and let's hang on to this a large part lecture because i'm going to apply this various drama please make sure that i have a correct yes i do this is independent of the mass of the object even though you say it isn't and here you will see that in all cases when we calculate moment of inertia about forty that there's always a mass of pure for the mass real disappears you will see very shortly i have four pics here and they all have different moment of inertia bilal going to vote it about an axis perpendicular for the blackboard so to speak and we're going to massage each one of them to predict their periods let's first go to the wrong three first-order rob we have the role here this is born people and here is the center of the road raut has mathram and has length l so we have here angie i don't have to worry about it anymore i simply go to this equation and i want to know what the purely this ralph is also may be wrong all i have to know now is one of the moment of inertia about p and i know already that being that he'd creation because one hard for the moment of inertia also mention about point p i have to apply now the pro-active spirit which also have to do during the exam which says it is the moment of the militia of rotation about the center of mass which in our case ec actually have to be parallel code is this axis perpendicular to the blackboard these axis perpendicular to the blackboard plus the mass of the world times the distance between p_n_c_ script and businessmen squared but it is being spread and the script is one quarter health career what is the moment of inertia for rotational paroled bumps this axes look that up in the table i'd happen to remember it now because i'm lecturing ada wong two months from now i will have to go so i remember another that is one twelve m l spread close one-quarter and health credit uh... would become to bomb group and helped create and sort of period g become two point times the scripts of his moment of inertia which is the one third and mel script app divided by b and g and b is one half hour with this trauma trick to one-half and g note is indeed as i anticipated always luger m also who's one hell here until you get to fight and escreveu of two-thirds l divided by g so that is the period that we predict products right that under here with regard to compare them shortly but it is to find trying to screw of two-thirds l divided by jim the wrote that we have here is designed in such a way that the periods is very close to one second that was our goal g as close as we can get it to one point zero zero seconds so if you're substituting this equation t called wong you'll find at the length of this rolled if it is really good looking at the very end should be about thirty seven point two senate leaders so we did the best we can when we make this world there's always an uncertainty of course how he drilled the holes and revealed the holes so i would say the belly that week actually achieved east thirty seven point two quote it was an uncertainty over about three millimeters so all point three sentiment that's what we have that is uh... a narrow one part in three hundred seventy let's make that ron got off the two one percent at work in the leg implemented on the desk where wrote the one-percent ever become scoff at the center soar the periods that i'd end up would predict is about one point zero zero zero was half a percent so that is all points seconds so that ben has one half percent visit might predict victory and so we're going to make and also nations often observed ulcerations we're going to get a number my reaction times no mud not much better than no type of tens of a second so we're going to get a number there we divide is number by ten and so we can always calculates the period and and then we get a much improved their work over hundreds of a second because we will divide this number by ten but it's also going to be divided by ten akikko close were able to get this food at one point zero seconds views the world turned is on the timeline also at the royal and i will start its when it stops somewhere nine points well within the prediction nine point nine two and so this becomes all points nine was a minus point don't want you see that as well within the prediction that i made now only four blacks were designed in such a way that they have exactly the same period but one sec annul comes the question hollow do the dimensions relate to each other in order kept in period but one sec so let's not calculates the period for the artistry objects ordering for the disk and for the pen let's start with the pendulum the pendulum send-off mass is here here is point pete i get dependent lengths for the royal use to get on the right record there sould be equals alpha that's the separation of the p_n_c_ the motive inertia about point b is very easy now this is no mass masses all here mass capital m but it must be and on health career so i go to this equation so i ask what is the periods of the pendulum and you're not surprised that defines coupon and is critical that will fit in we've seen that before but of course it also comes out if you do it in this more complicated rate so here we get to part i'm to screw let's not do the the ring is in the ring they did about forty people yet here is the center of mass right now in the middle of no where and here's a sample of my house and sold the distance b is the radius of the ring so we have to calculate moment of inertia about point b again we listed after he was not a pro-active here and that is the moment of leadership rotation bob this axes two point eighty percent of mass that is an *** created arteries of the ring all the masses at the circumference or in addition to our from the center of mass and then we have to add to that according to the collective theorem the mask on this distance script and efficiencies or so you get a more screen so it is to win are spread so what is the period open also mation i love pete that is to m our script divided by the m_g_ views or or mt and it was one r i lose an m and we've seen this before either write this back to number twenty one i remember everything by election on the twenty you've got photoperiod now is the same as the pentagon with links to our and so here we have to fight i'm to scramble clark provided by geometry comparison very shortly i just want to finish the malt and i know i would like to do the disk but not least what did they ask over half of the most calculates the moment opener show this was very close to you the problem you have only exam you have the disk salted disk uses point p because the center of mass but now it's all that so we can be these are separation between c and d and the moment of inertia for rotation about point pete is not a moment of ownership for attention about the center of mass which you will go up in the cable in the case of your time it was given on the front cover one-half and our script that's the moment of a new ship rotation about the central maps now we have to ad and be screwed and vehicles are so we have to add and mark twain so we find freehoff and north korea so what is the periods oscillation find three house and more scripts divided by d and g these are or and that equals to fight app and his crew and who's my and i thought was and wasn't are i get here three haha departed biji so let's write that down here so we have to fight i'm screw freehoff are by the budget so we have the mole spore there and so we can now make meaningful comparison we want to period for me to say so we can hang on to those numbers so we don't need this anymore they want the period to be the same we already have established that the period or the royalties close to the one seconds so we're not going to mail them anymore we just want to compare them wedding date they have the same period of automation bite making them also making unison but we want to know what the relative ratios are visa mentions the way we decided so let's first go to the welfare make to compare something to ralph and the pendulum so if you look at the wrong and we use the pendulum as our spend it with lately royal then you see the period will be the same it one-and-a-half art is little help also review robbed if two-thirds capital l is little else so for the road two-thirds l nichols little l so capital l it's gonna happen has to be exactly one-and-a-half the legs of dependence of this length has to be exactly one-and-a-half times this length of the central mast it's it's essential but don't you and that's what we try to the best we could let's now go to the ring for rain have the same period one seconds to our has to be the same as the length of the so too are equal l so we can put in here for l this has to be one-and-a-half times l in here this now which is to our has to be failing to achieve at this link here at the same at the demagoguery moldable albans and so now we go for the desk so now we require that three house are one-and-a-half art we won't have to be l so we want our b_-twelve divided by three so we wanted diameter to our to be four l divided by three and when you look at the desk hard to see that it is exactly four third street you can see it is longer than the pendulum one-third longer but it is not as long as the world because the law royalties one-and-a-half times the length of the pan and so we can all complete that picture and we have now the diameter of the year now four thirty and so we can do it all we can play with whom we can also make them simultaneously and deceit how well they track each other uh... there's no sense in getting into that mentions the world what thirty-seven point too so let me just let me write that down because we calculated that so this will thirty-seven point two centimeters and i think that translates into for the pendulum twenty four for dates enemies forty others and leave it up to you calculate dimensions yes twenty-five twenty four point eight correct timing is not useful anymore let's just see how these two go together all set them or let them go they go pretty much in unison if you're a blowing up of course you will see that is a difference you can ever make an exactly the same but that wreckage of the nicely we can also use the roles and the disk trek into the beautiful bulls very close to point zero zero seconds and we can have the verses the pendulum and you see that wreckage of the very nicely but wait long enough and you will see that of course the periods will be different so that you might last word on physical pendants but you may see began on the final not needed you can almost compliment okay i want to discuss now some other interesting automation again simple demonic and that is that correct in a you tube if you see there if i have here which has every right to open on both sides everywhere the same cross-section and a politically in here uh... equilibrium thus like that and looked at the liquid has mass and identity role the area of the two eight and the legs opened eclipse l this is i'm going to offset it liquids and i wanted to see also it's i want to see working calculates the period of the oscillation the total mass of the liquid that i have total miles is the volume which is every hein command control i'm going to offset it so that this is higher over a distance so there's that is lower over a distance for this distant is also why things that malik without is here and then i really since and it will start to possibly when it starts to also late there comes a time that the liquids the whole decreed is going to sloshed back and forth and so everywhere in the the velocity of any moment in time will be the same because the cross-section is not changing the same everywhere there's a certain velocity here p and it's the same as the velocity here as a velocity there as the velocity there and that of course it is why dot that's the preservative of that position here i'm going to right down the conservation of total energy mechanical energy i'd assume that there is no energy loss although the probably some friction inside the liquid will probably generate some dates and that will cost and benefit you will see that we have a duty experiment but now i will assume but that's not the case so what now is then the total energy of the system headed to some of the kinetic energy plus the potential energy if we assume that that's constants we will be able to find superior topical solution very shortly as you'll see the kinetic energy although they create is easy that is what happened times the velocity square in the velocity we agreed is white dots great not a potential energy i call the potential energy here i call that you'll of course zero when the liquids standing here and the liquid is standing here i call that potential energy zero the mass that is now above this level here i called adult a m and our plan the area times lifetimes role and how much master as he was taken away from here and sport here how much work do you have to do to take this liquids unported there but it's the same way you take this victory and put it here and when you bring a liquid rituals bear here then you have moved it up over a distance wine and sort of gravitational potential energy increases by delta and the the amount of last year bungee underage and eighties why and g_-eight remember that's increasing potential energy so i'm more when i'm out of my house which is delta n unwanted over the distance i bring it here but that makes no difference of course and so this is well total energy and this is not a constant so i'm going to substitute in there the a ellen role so i could one-half velocity script was eighty jeans and i get a widespread equals a constant because of the why here and have a wider we've done this before this is the conservation of energy and in order to find two periods off the simple monocle solution we take the time derivative by the way before we do that uh... this was dealt a m than aid right ear a hero yep that's it so really great we neutral and we continue with what we have so we're going to take the the renegade versus time of this equation because he won't call l the tool pops out and it will become too long so i get to why dot that i find it a little so i got why double dot greg apology to pops out you comes to why and then i get chain rule why dot and that you called i lose my white off because they have wide open boast rooms this to heat up this too and so i find that why double dot floss fuji divided by l time tomorrow equal zero and backed walls my goal this is clearly a simple harmonic oscillation because this is a constant and it will also lakes deliberate white people's wimax uh... to co-sign i'd only gotten was file it with the angular frequency which is directly related to the periods omega angular frequency equals this crew apology over l until two periods will be to fight finds this crew well old fuji but this is the periods also made it is the period that you would have had for we're booked a form of pendulum if the length of the clinton was a little over two not at all of these mogul intuitive we our set up here i have to know what analysts and that is not so easy because of this radius here if i measure element outside it substantially larger than on the inside you're a don't think it's a bit different the did you notice in nine seventy two different three outside than inside if i take the average value between the two if i take average i find seventy two centimeters and i could be off by one if i use this number four allele and i substituted in this equation that i find i predicted periods which is one point two zero four and because of this error that i have a while that would give me uncertainty over bonn's app opentable wild seconds however before we start measuring it than double-dip can also nations to get a reasonable accurate results i want to warn you i make a prediction that the period we will measure will probably be larger than this and i can think of two reasons forces that the deadly obviously liquid will be you which you will see how quickly events in the past we've never taken dumping into account and we want to live in a tu wong with the done thing has the effect on the king the period longer we've always ignore that and in most of the demonstrations that we did like just now that was acceptable they'll be acceptable for the liquid thought it was the second point that i want you to think about is it correct that i take the beverage namely the average value between the outer any intellect i don't think it is i want you to think about why that is not correct looked carefully where that l comes into my differential equation and you will probably come up with the right answer and i claimed that the actual al that we should have taken is a little bit larger i don't know how much larger but it's a little bit larger and so that will also make you observe periods become larger than the predicted well so i'm not too optimistic that we will go and get this the way we want to get it but that's good because that's where the physics lies with you see there other factors that have to be taken into account alter on this long has it all now he wrote when they could completely dark in the classroom good you going to see all the ways you can see the liquids wish you good luck with now or a strategic reasons too okay it zero so let me try to get busy the swing like shrink it down so enormously that i really want to get a very large swing that's nice lebed's twelve point one eight moped a little bit of lights twelve point one eight soldier canteen observed itself point one eight okay my reaction time at all point one so ttyl observed is one point let's make it through was a minor dole point o_ one second all that's not bad it's actually there's overlap if you add this one here you get one to one and if you shipwrecked if you could also want one so it's not that hike speculate to be a little higher but close enough to be happy think about by l should have been taken a bit wiser now well more very interesting ulceration torsional pendulum so why are there to help me to steel wire was hanging something on the bottom which we're going to offset and that is going to also make back and forth at school torsional pendulum and we're going to calculate period of operational for look at them and they have wonderful properties they're nowhere like aspirin like the one that mention all spring remember the one that mention will spring that we had the period which was independent albee and that you well within reason of course to make impeaching too large and you get permanent deformation spring we never have to make any small angle approximation rizvi spring as we have to do with the pendulum here is the pendulum the torsional pendulum is a bar and it is the rate even as a way to you i'll tell you more about that later hanging from the ceiling has a certain length l this is point p we're going to twist it and then we don't want to let its postulates barton horizontal plane when you look from above it with you to bar here pointy ears and then we can offset it over an angle fade out and then it will also laid back in force the port were quick to point pete no very similar to what we have to resist playing we have a minus sign done that illustrates that it is restoring instead over a okay now we have cap rock which is what we call the portion will spring constants and now we have an angle which we call so we generate report which is proportional forty-eight very similar to the many a spring whereby the generator force which proportional to the linear displacement now you generate torque than a proportional to the angle and at least the moment of inertia about point pete finds all fine and also i stayed up double dot so we're going to get that probable outcome plus the cap rock find feder divided biography equal zero couple by the way proportional constants so we have a different integration clear that you're going to see eight simple monocle solution this is a constant so you're going to get fade out equal fatal maximum i had to co-sign omit accomplice wife getting boring isn't it angular frequency and in the frequency and and get a free country this grand of kappa divided by at the moment of inertia about twenty and therefore to periods which is approved by divided by the media nichols to park of the screen woman of inertia about point b divided by a kappa how about kappa private function of the cross-sectional area year eighty and the length l and it's also functional thought kind of material you have let me ask you one night will makes a big difference that's very intuitive of course remember that you know your lecture when we stressed hawaiian to the point that it was breaking we dealt with jones most of us we had a wire and what a mass hanging at the end of the wire and we discussed beat vertical oscillation stress it and let it go and then we will get an oscillation like a spring and that spring constant that we found them was months of young small jelous obstacle section where we need your divided by toning and that was kind of crazy mistake of the bar step for this the longer the wire the best if it is thought of something very similar here but i don't want to go into the details of exactly how did the right here cap rock it's a little bit more complicated but indeed the same is true if you make the wider thicker then kappa will go up and if you make the why you won't work kepada go down that's immediately obvious if you have a very short lol and you try to twist that brought the plant at the top of your twisted and it's very short you would need a tremendous pork for ten degrees if you make the why a hundred meters long and you want to push the ten degree fixed nothing you can immediately see that of course value for cap on the torsional constant is a function of the legs will go down when the length goes up we have a while you were just two-and-a-half meters long and the thickness of the wire the diameter according to the manufacturer to piano string uh... the thickness of twenty five thousand orphanage and if i calculate kappa to the best of my ability i find that cap rock to be very close before constantly reminded sport no two meters radiant and so all we have to do now is to calculate the moment of the militia of the system and then we can predict what the periods of the spending is going to be which is not my gold you'll see my goal is going to be different look at the bar and look at the wire the wireless located at the moment of inertia wrote a fifty-point p of the wire is very close to zero remember it is proportional to our square you can forget about that almost all moment of inertia is indices i'd a blow-up that system for u here it is you'll see it there ahead on both sides as two hundred grams diso point two kilograms and it has here opelika commitment and is mass is almost negligible and this this this year thirty centimetres and this distance thirty seven years go to a very good approximation the moment of inertia go rotation about that point the this rotation well be death mask andre discredit whatever map factories clear so too will be quite because i have two masses fiberglass activities script and that is about all point o_ three six kilogram peters square and so when i use that into all week regime so i don't know what cap rock is at least i have a reasonable ideal cabarets and i know what i hope this is really do almost exclusively determined by the crossbar i will find them using that equation at the periods very close to sixty seconds my goal is not still prove to you that it is close to sixty sec my goal is to show you that's what this time mentioned which is very thin and very long recon make that angle theta maximum this angle amazingly large you're not talking about ten degrees wat thirty degrees we can go much further and what i want to test the humorous how far we candle is one of the great things in life for you and for me to challenge how far can you go and get away with it but comes a time that if we make the angle too large that we'd probably be a form the white it will not come back to its original position the same happens which is a point if you take a sprain it is true that the period of also macias independent of the amplitude but only up to a point if you go to florida deluxe wall no longer hold picture before it prominently then of course the period will become a function off the extension of the same is true here so if we get twisted to watch up then of course we will permanently before myths and then the periods will not be independent of state a maximum having said that i would like to start asking you for advice what kind of this direction so we start with what do you think is reasonable without total torture and that will right down the times so we're not really interested in testing the sixty seconds but we would water like to compare the various angles that we get into one of the first one we will be will try any idea thirty degrees but dries you were kunal man the first brought you want to expire bottom line i'm reading i'm ready to go walking wrote rotation ok you think it's nothing to speak that's for you right well i would like to go straight up maximum by three hundred sixty degrees total part and measure depute in fact remember the peoria takes a minute and it's not necessary we can measure papa periods we wait until the pentium stops every resident time until it stops again the top of periods likely to spring if it stops here and stop their the top of period so measure hopper periods i don't know what my reaction time is going to be maybe not a tenth of a second because the moment that it stopped is not so well defined you will see i'm just guessing public larger than one-tenth of the second let's give it a shot let's try first three hundred sixty degrees dated black in the civil red so will rotate its one rotation speculate walls yousendit during the sixty degrees okinawa for athletic colin wait until it stops always do that then i start the time never that stops again i'll stop the time that we have hopkins but first we don't think very slowly very gently fertig roughly thirty second sport operative issue so you'll see now that it's nordica librium again every one of the bump rotation and so spektr equilibrium and now it's going to stop very shortly someone stops i want to start prime okay law goes back relating to the folks again is as powerful twenty eight point seven five twenty-eight pointing what are you know free rotations five rotations three uh... we favor of three who was responsible for permanent damage to the wire accept responsibility life raise a lot during his six bartamannb i can start because it will take a while before it stops three rotation for as we have to be sure that it is moral is back equilibrium which is always difficult thing because it's so slow got close enough okay street illegal clockwise or counter clockwise should make no difference i went this way for us so we go back sleep with that you know why okay let's go i'll start a time when stopped we have some time expired well relocated six three times and it will locate six times back before it's false hope you realize that would like to make thank you working that out thirty rotate it three times and then let it go at those first three times back then it's a career and they want three times up again before it stops look mode is it going much faster now that the time that's the whole thing should be very close to that number again twenty eight point five bad so they know go all the way what he wanted to know break though i order then rotations you'd love to see that right logo like matt ten rotations that amazing how much factually gulch just still pretty set must make sure that i have my equilibrium this was not equipping them i know it somewhere here no it wasn't equilibrium and this is it dendrites well poorer six seven date line ten who are here let it go we'll see what happens when it stopped i'll start the time thank you thank you for pointing that out look how fast it's going to be easily going wacko has to do all that in thirty seconds would not have to go back to it stopping estimate twenty locations now twenty rotations in thirty seconds them back to equilibrium tend to come to a halt they're going to be your thanksgiving farewell demonstration twenty nine point two fantastic okay have a good thanksgiving