Wikipedia:Reference desk/Archives/Science/2009 April 10

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April 10

can we get any "object" to anywhere near relativistic speeds?

can we get any "object" with significant mass (I don't really mean single particles, but it doesn't have to be a big object) to anywhere near relativistic speeds, either here or in space? 94.27.161.254 (talk) 09:55, 10 April 2009 (UTC)

Well, this is a somewhat vague question since there isn't any solid definition of "relativistic speed" or "significant mass" - but taking it in the spirit in which it's asked - no, we have never gotten anything weighing (say) at least a microgram up to (say) 90% of the speed of light. Single particles - yes, certainly. SteveBaker (talk) 10:18, 10 April 2009 (UTC)

I'm not expert on this subject so I'm prepared for one of our resident physists to blow this idea away, but I read somewhere that it takes between the MeV or TeV range of energy to accelerate one proton to a decent fraction of the speed of light, which is a lot of energy for just one particle. Add in a few more MeV/TeV for a neutron and another MeV or so for the electron, and for the most basic of atoms you're starting to rack up a decent amount of energy. Let's say, for example, it'd take 3.204 × 10^-7 joules to accelerate a hydrogen atom to near the speed of light. Not much? Except we have to remember to get a sense of scale, an atom is nothing in terms of an object. To get one mole of atoms to near the speed of light, by the previous figure which I'm not convinced is anywhere near accurate, we'd need 1.92 x 10^17 J, which is near about the amount of energy that strikes the Earth each second from the Sun... that's a lot of energy. —Cyclonenim | Chat  11:38, 10 April 2009 (UTC)

It looks like you just made those numbers up, am I right? We can calculate this, you know. In units in which the speed of light equals 1, the relativistic energy of an object of (rest) mass m at velocity v is ${\frac {m}{\sqrt {1-v^{2}}}}$, but m of that is its rest energy, so the extra energy to be supplied is $m\left({\frac {1}{\sqrt {1-v^{2}}}}-1\right)$. At v=0.9, the value Steve was considering, that's about 1.29m. For reference, the Hiroshima bomb liberated about one gram's worth of energy.

So for a bomb and a third you can get a one-gram object up to 90% of the speed of light, if you can figure out how to turn all that energy into KE of the object. That is a lot of energy, but it's about three orders of magnitude less than you came up with. --Trovatore (talk) 01:02, 11 April 2009 (UTC)

I believe that the fastest object created by man is the Voyager 1 spacecraft which is currently traveling at 38,350 mph. Light, by contrast, travels at about 11,176,943 mph. So Voyager 1 is at about 0.34% of the speed of light. You might be interested in Project Orion for some theoretical possibilities. A Quest For Knowledge (talk) 12:22, 10 April 2009 (UTC)

That should be 670,616,629 mph, which would put Voyager I at about .0057% that of light – equivalent to a true snail's pace (2 mm/s) on a superhighway (120 km/h). -- Tcncv (talk) 00:39, 11 April 2009 (UTC)

That would fit my def of "relativistic speeds", that is, speeds high enough that relativistic effects must be considered. At that velocity I imagine time dilation needed to be taken into account when using an on-board timer to tell it to come on and take pics of a passing planet or moon. StuRat (talk) 13:29, 10 April 2009 (UTC)

The LHC will be able to accelerate lead nuclei to just shy of the speed of light. I think that's pretty big. 163.1.176.253 (talk) 12:37, 10 April 2009 (UTC)

Accelerating a lead nucleus would still be in the order of a microjoule or so, and the LHC plans to use quite a few in order to achieve collisions (otherwise atoms will just miss each other), so the energy requirements soon add up. —Cyclonenim | Chat  12:42, 10 April 2009 (UTC)

While it certainly has not been done, I wouldn't be so quick to say it cannot be done. An orion ship ( http://en.wikipedia.org/wiki/Project_Orion_(nuclear_propulsion) ) could reach 10% lightspeed. Beam-powered solar sails ( http://en.wikipedia.org/wiki/Solar_sail ) could reach "a significant fraction of the speed of light" —Preceding unsigned comment added by 81.11.182.18 (talk) 12:34, 10 April 2009 (UTC)

You might also be interested in NASA's antimatter spaceship.[1]. A Quest For Knowledge (talk) 13:13, 10 April 2009 (UTC)

If I remember correctly, astronauts aboard the space shuttle have been calculated to experience time about 1x10^-9 seconds faster than those on Earth. ~AH1^(TCU) 17:14, 10 April 2009 (UTC)

Ornithosis agent

I ran across this article since it was uncategorized. I don't know anything about the topic, but as far as I can tell, ornithosis is any infection spread by birds. Orni- or Ornitho- is Greek for bird, and -osis is Greek for disease. So I assume an ornithosis agent is any pathogen present in birds. The article mentions Rickettsia. I'm unsure if the Ornithosis agent article should be deleted, redirected to Ornithosis (actually to Psittacosis since Ornithosis currently redirects there), or Chlamydophila pneumoniae, or Chlamydophila psittaci, or perhaps moved to Wiktionary. There are several hits for "ornithosis agent" on Google, Google Books, and Google Scholar. But maybe it's more suited to a medical dictionary? --Pixelface (talk) 11:19, 10 April 2009 (UTC)

This is more a matter for the WP:HELPDESK than for us. But in general, if there is more to be said on a subject than a mere definition of the term (and that seems likely in this case) - then if it's notable - then we should have an article about it. The word "Ornithosis" has 70,000+ google hits and "Ornithosis Agent" gets about 16,000 - so neither is something someone just made up. I guess the big question is whether it's just a synonym for some other more common term - in which case merging the articles together and turning the less commonly used into a REDIRECT page would make sense. SteveBaker (talk) 11:26, 10 April 2009 (UTC)

Thanks for your prompt reply. I considered asking at WT:MEDICINE, but the top of that page mentioned this page. --Pixelface (talk) 12:41, 10 April 2009 (UTC)

Merging the sentence from ornithosis agent into ornithosis would be the best plan, and then create a redirect to ornithosis from ornithosis agent. Regards. —Cyclonenim | Chat  12:45, 10 April 2009 (UTC)

I would not merge them just yet, as we have a bigger problem. The thing is, there is a microorganism Chlamydophila psittaci, which causes bacterial ornithosis (psittacosis), but it is not a Rickettsia. So, either ornithosis agent article is simply wrong, or it refers to a different microorganism. Would please someone who is either a MD or a certified vet sort this one out? To make things worse, there is also an ornithosis virus, which we do not currently have an article about; but we should. Any ideas? --Dr Dima (talk) 18:54, 10 April 2009 (UTC)

I replaced the contents of "Ornithosis agent" with a redirect to Chlamydophila psittaci. --NorwegianBlue ^talk 11:50, 11 April 2009 (UTC)

Limits of curving a ball

In particular I have in mind the limits of curving a football (soccer ball), that is hitting it in such a way that it travels along a curve rather than a straight line (like this). To be honest, I don't really know the physics of the motion, but I believe it is possibly to do with high / low air pressure on one side? I believe the primary variables would be the speed the player strikes the ball, the point at which they strike the ball, and the size and shape of the area of the ball that their foot meets. What is the theoretical maximum curve a human could impart on a football assuming normal environmental conditions, a modern football and so on? To phrase the question more concretely, if a ball was positioned 25 metres from a vertical pole and a straight line is drawn between these positions, what is the maximum deviation from the straight line the ball could have when struck in an optimal way such that it still collides with the pole. --80.6.184.236 (talk) 13:48, 10 April 2009 (UTC)

The curve is due to spin, the Magnus effect. So you really need to figure out how rough a shoe one has (friction during the kick to spin the ball), how off-center you can kick it while still being able to control its direction, etc. There's another oddity from the video though...you're seeing a 2D projection of 3D motion, so there are all sorts of perspective problems (for example, parallax) that can make it hard to see the actual trajectory. DMacks (talk) 14:06, 10 April 2009 (UTC)

Footballers are capable of giving significant curl or dip to a ball with little or no spin. This is not a well understood area of science, due to the complex nature of the problem. There are factors which are often ignored (such as the non-rigid nature of the ball, and its non uniformly frictive surface) that are simplified out which are now believe to be major factors. I have been told (whether or not reliably I couldn't tell you) that oscillations in the shape of the ball caused by the initial impact of the players foot play a large part in curl. Elocute (talk) 21:53, 11 April 2009 (UTC)

I'd favor an experimental approach here (get a professional soccer player and have him give it as much spin as possible, then measure the results). Calculating it mathematically wouldn't work well because so many factors are unknown, like the amount of friction between the shoe and the ball. StuRat (talk) 15:48, 10 April 2009 (UTC)

Dashboards

Not to sure where to post this, feel free to move it (engineering? law?)... Does anyone know where I can find a precise list of the dashboard instruments (and tell-tales) required for a car to be road-legal in the U.K. ? A client wants a road-legal prototype ASAP, and I need to know what has to be finished, but I can't find this anywhere. I suppose the speedo is obligatory (in mph only or in kmh too?), but I'm not sure about the rest. If anyone has a Caterham or similar, could they tell me what they've got (I'm guessing it's pretty close to the minimum required)? Thanks! yandman 15:09, 10 April 2009 (UTC)

Most of the info you require is contained in ECE Regulation 121. The rest can be picked up from ECE R39 (speedometers) and R48 (installation of lighting and light-signalling devices). —Scheinwerfermann ^T·_C03:28, 12 April 2009 (UTC)

Ouch. Someone's going to be doing overtime... Thanks! yandman 08:39, 12 April 2009 (UTC)

are flywheels any easier in space?

Note: flywheels are inherently mechanical devices akin to hard drives in the sense that they enclose spinning platters -- however in flywheels these platters spin in order to store energy -- thus the flywheel is a kind of "battery". This question concerns the design challenges of flywheels in space, namely whether designing flywheels becomes easier when they are to be in space. The question is purely theoretical and is not concerned with the economies of actually getting the (simplified?) design of the flywheel into space, or charging it with power! 79.122.65.253 (talk) 22:04, 10 April 2009 (UTC)

the question

Would a giant flywheel be much easier to implement in space for the following two reasons:

1. Space is a natural vacuum
2. The motor does not have to bear the weight of suspending the flywheel

I'm thinking: why not use two huge flywheels attached to your ship in a tandem rotor configuration (so your base isn't spinning wildly in the opposite direction)?

b. serial motors?

additionally what's wrong with this reasoning:

• since you don't have to support any weight, couldn't you weakly couple (no load) a bunch of cheap, small motors serially to get the full rpm?
• This insulates the base from the huge speeds of the flywheel and means you don't have to use such powerful motors or worry about:
  • the problem of friction at high speed
  • the problem of trying to couple electrically at high speed (in other cases the flywheels might not need any electricity, but in this case the rest of the motors do)
  • the problems such as eddy currents that emerge at high speed

For example:

O------[ship]------O

(where the O's are your two flywheels and - are the weakly coupled [not load bearing] independent motors)

In this example here are the relative RPM's at each motor once the flywheels have fully powered up:

O5-5-5-5-5-5-[ship]-5-5-5-5-5-5O

But here are the RPM's relative to the ship:

O15625-3125-625-125-25-5-[ship]-5-25-125-625-3125-15625O

As you can see, you now have a huge RPM at the edge (that you slowly built up to) but each motor just has to turn a maximum of 5 rpm relative to the one below it -- that means each motor can turn with a huge torque, for example to turn flywheels that are really really massive.

This is how I imagine it would spin up:

• First the innermost motors -[ship]-
• start sloooooooooooowly turning the static rest of the motors+flywheel O----- and -----O
• once the innermost have brought the rest up to 5 rpm (where they remain relative to it) ...
• the second pair of motors --[ship]--
• starts turning the static rest of the motors+flywheel O---- and ----O
• etc etc

In this way any flywheel could be easily brought to the speed where it nearly shatters. It is also not hard to transfer electricity from one flywheel to the next, if it's only moving at 5 rpm! you could even use just a simple "brush" connector (not sure what to call it) it is a piece of cake.

I figure since friction is a function of speed, the relatively really small speeds between each section means the MAIN, main engineering challenge with flywheels (other than the load bearing part), ie reducing friction, becomes trivial. Also: you don't have to worry about trying to reduce the mass of the motors (making them lightweight), since they are also part of the flywheel and moving them isn't a "waste".

Are my logic, physics, and mechanics right, or am I missing something? 79.122.65.253 (talk) 16:03, 10 April 2009 (UTC)

We need to back up here first and figure out what your overall goal is. A flywheel is normally a device for storing mechanical energy. Is that really what you mean here ? If so, what is the source of the mechanical energy that you wish to store ? I'm thinking that perhaps you're talking about rotating part of the space station as a way of achieving artificial gravity. Is that what you mean ? StuRat (talk) 17:59, 10 April 2009 (UTC)

If you are talking about using flywheels to store energy, then bear in mind that moving large masses into space from Earth would be extremely expensive. If the flywheel could be made of something we already need to bring up, like maybe water tanks, then that would be another matter, although we then have the problem of how to access the contents while in motion. Another option might be to construct the flywheels of objects found in space. StuRat (talk) 18:03, 10 April 2009 (UTC)

The question is quite specific, as I gave in the title: "are flywheels any easier in space?" (ie easier to create under space conditions). As for uses, I believe that the flywheel article lists the uses for a flywheel, which all revolve around (heheh) storing energy. 79.122.65.253 (talk) 21:09, 10 April 2009 (UTC)

I've made this thread small and put a note at the top of the question explaining that a flywheel is for storing energy and that I was asking whether designing one is easier in space (given the reasoning in the rest of my original question). I hope you also feel the explanation now at top makes this change appropriate. 79.122.65.253 (talk) 22:01, 10 April 2009 (UTC)

Tether propulsion, Tether satellite Non-rocket spacelaunch There also was an idea in the eighties that looked somewhat like a shuriken and would have spun to catch and released metal balls like a space Jai alai game. Can't find the article anymore. (If someone knows what magazine it was in plse. let me know.) 76.97.245.5 (talk) 22:27, 10 April 2009 (UTC)

thank you for the links, which I looked at, but as near as I can tell none of them have anything to do with my question! 79.122.65.253 (talk) 23:12, 10 April 2009 (UTC)

Seeing as your ship is not anchored in space, by conservation of momentum it will gain rotational energy too. And since it is much closer to the axis centre, it will have to gain much much more rotational speed than your flywheels (Assuming it is of similar density). (left by 92.22.160.99)

maybe I didn't make it clear but like most tandem rotor configurations the two rotors would be spinning in opposite directions, for 0 net spin at the ship! 79.122.103.33 (talk) 22:15, 11 April 2009 (UTC)

There's a little problem with the math for your serial motors. If each motor is turning at 5 rpm relative to the one adjacent, there will be a linear progression (5 - 10 - 15 - 20 rpm) down the line, not geometric one (5 - 25 - 125 - 625).

Aside from that, there's no obvious reason why you couldn't stack motors in this fashion. Do bear in mind, however, that the same concerns regarding centrifugal force and dynamic instability apply to this set of chained, rapidly rotating motors as apply to a flywheel. The high-speed motor stages are going to be rotating very rapidly and have tremendous angular momentum, even if they're not moving very fast relative to the adjacent motors. From what I understand, constructing high-speed motors that will drive a flywheel isn't a major hurdle, even here on Earth. (Vacuum and zero gee do may make life a bit easier, though.) Our big problem is building flywheels with sufficient mechanical strength to handle really fast spins without exploding, not getting them up to speed in the first place. (Laboratory folk will be aware – hopefully through instruction, rather than personal experience – of the incredible destructive power of a failing ultracentrifuge rotor: [2].) TenOfAllTrades(talk) 22:30, 11 April 2009 (UTC)

[edit conflict] Your notion that serial motors would have multiplicative rotation rates is false; they would be additive (as Ten said). If you want to reduce the speeds of the motors to reduce friction, well-lubricated gearing systems could spin the flywheels much faster than the rotors; no serialization required. Certainly it would make things easier to not have to have bearings that were load-bearing. --Tardis (talk) 22:33, 11 April 2009 (UTC)

Yeah - "serial motors" is a dumb idea. Motors are relatively big, heavy things - and getting power to a motor that is itself spinning is difficult without resorting to 'slip rings' and such which will dramatically increase friction - and will wear out in no time at these crazy RPM's. So use gearing to multiply up the RPM's and have either on large motor or several motors "in parallel" driving the main gear shaft. Also, the torque on the shaft of the first motor is the torque on the last motor multiplied by the number of motors - so any bearing friction in the last stage would be multiplied up and would require a gargantuan first motor and an only slightly less gargantuan second motor - and so on. We can make insanely large electric motors - and they are pretty efficient. There is really no reason to mess with serial motors. SteveBaker (talk) 15:48, 12 April 2009 (UTC)

Name that UK bird! Orange chest, black head.

I live in the Midlands of the UK and I just saw a bird in the garden that I had never seen before. It had a black head and a vibrantly orange chest but with black tail feathers and darkish wings. It had a black beak which I couldn't tell the shape of (hardly distinguishable from the head, so I'm thinking small). It looked a lot like a parrot and was curling its head down in a way I'd call parrot-esque, size wise was smaller than a pigeon, much bigger than the robins and blue tits about. First impressions was of a small parrot, sadly no photos yet. Any ideas? MedicRoo (talk) 18:40, 10 April 2009 (UTC)

Turns out it was a bullfinch.MedicRoo (talk) 20:20, 10 April 2009 (UTC)

Early cyclotron-thing

Resolved

I can't seem to find a picture I know I've seen on Wikipedia of an early particle accelerator or isotope separator or something. It is roughly spherical, made of brass or copper, and looks a bit like the helmet at right. The photo has it sitting as a historical objet d'art in a courtyard at a place like CERN or Fermilab. I just can't seem to remember where. Anyone know? Thanks.

It looked a bit like this

Is it possible that you're thinking of Henry Moore's Atomic Energy sculpture located at site where Enrico Fermi conducted the first self-sustaining nuclear chain reaction? A Quest For Knowledge (talk) 20:37, 10 April 2009 (UTC)

Nice try!, but no, it really was a bona fide scientific instrument that had been retired but not scrapped due to historical importance or its industrial beauty. --Sean 21:35, 10 April 2009 (UTC)

Wilson cloud chamber? They look sort of like that. Though not in the Wikipedia picture. --98.217.14.211 (talk) 23:29, 10 April 2009 (UTC)

A Van de Graaf generator? Or a Cockcroft-Walton voltage multiplier? Rmhermen (talk) 23:51, 10 April 2009 (UTC)

Got it, I think. See Microcosm (CERN), which has this image, a resonator from the Large Electron-Positron Collider. Mikenorton (talk) 10:47, 11 April 2009 (UTC)

YES! Thanks so much, that was driving me crazy! --Sean 13:47, 11 April 2009 (UTC)

Need WP:RS on a Common Misconception about Neutron Bombs

I'm currently working on our article List of common misconceptions. This article is/was horribly unsourced and I'm trying to add cites to some of the misconceptions. Although I've been able to find cites for all of the ones that I've worked on so far, I'm having difficulty regarding the one about neutron bombs.

Specifically, the article makes the following claim:

"*It is commonly believed that Neutron bombs are nuclear weapons whose blasts exclusively affect living tissue. This is not true. There is still some heat and blast energy but at only a fraction of the intensity of a conventional thermonuclear warhead."

I believe that the claim that this is a popular misconception is essentially true: that people mistakenly believe that neutron bombs only kill people and leave all the buildings standing. But I can't seem to find a WP:RS that actually says such a thing.

The problem is compounded by the fact that the source needs to say that this is a common misconception (or words to that effect). A Quest For Knowledge (talk) 20:27, 10 April 2009 (UTC)

The trouble is that the misconception is somewhat backed up by the facts. As your article says "This is still some heat and blast energy but only at a fraction of the intensity..." - which means that providing you make an air burst at sufficient altitude that the heat and blast doesn't destroy buildings, it could well be that living beings are the only significant casualties - and the buildings remain intact. I have no clue to the degree to which that is true - but there is certainly room there for this not to be exactly a misconception. I mean, sure - if you do a ground-burst, it's going to level a lot of buildings and set fire to a bunch more - but it's not beyond the realms of possibility that it could be triggered such as to annihilate the populations and leave buildings and other infrastructure alone. SteveBaker (talk) 22:33, 10 April 2009 (UTC)

Well, it's not my article. I just started working on it a week or two ago. Anyway, the first part I think is the misconception "It is commonly believed [by whom?] that Neutron bombs are nuclear weapons whose blasts exclusively affect living tissue.". What follows is the correction: "This is not true. There is still some heat and blast energy but at only a fraction of the intensity of a conventional thermonuclear warhead.". To be honest, the second half may not be phrased the best. My understanding is that the blast is still larger than a conventional weapon, and I might rephrase it to say that. I think the point is that there is a false impression that that only people are killed and all building are left standing. But I see what you mean about if the blast is sufficiently high. If this item is factually incorrect, I can remove it from the article. A Quest For Knowledge (talk) 22:45, 10 April 2009 (UTC)

We have an article Neutron bomb in popular culture, which has many verifiable facts, such as a given quote in Repo Man. You could rephrase from the survey-requiring "it is commonly believed ..." to the easily sourced "it is commonly depicted in popular culture ...". --Sean 22:37, 10 April 2009 (UTC)

AQFK, while you are at it, may I also humbly suggest that the points given in the article be presented in a structured way to avoid confusion? For example, one of the points says "Snapping or cracking one's knuckles does not cause arthritis.[59]". Now there will be a confusion for the reader whether the statement is a myth (i.e. snapping the knuckles does cause arthritis) or whether the statement is a myth buster (i.e. snapping the knuckles does not cause arthritis). May be we can add the word Myth, state the myth and then explain why it is a myth ? - WikiCheng | Talk 05:55, 11 April 2009 (UTC)

I'll keep that in mind, but feel free to be bold. A Quest For Knowledge (talk) 00:37, 13 April 2009 (UTC)

Rolling hemispheres and moments of inertia

Here's a question that appeared on the math reference desk and didn't receive any responses. I wanted to know the myself and thought it might have more luck here. --Bowlhover (talk) 20:43, 10 April 2009 (UTC)

Hi there - I'm working through a bit of mechanics revision and have been looking at some of my uni past exam papers from a good few years back when the syllabus was different, so I'm not familiar with a lot of the material covered - it's not strictly relevant to my current course but I've found myself stuck on this question about hemispheres and it's bugging me, relevant or not. Could someone please give me a little help with how to progress? Thanks a lot.

i) Use the parallel axis theorem to calculate the moment of inertia of a uniform hemisphere of mass 'M' and radius 'a' about an axis through its centre of mass and parallel to the base (the centre of mass is located at a distance of 3a/8 from the flat face of the hemisphere).

Well the moment of inertia of the hemisphere about the axis which would be through the centre of mass of a full sphere is ${\frac {2ma^{2}}{5}}$, as it is for a sphere, so by the parallel axes theorem, $I_{c.o.m}+M({\frac {3a}{8}})^{2}={\frac {2Ma^{2}}{5}}$ so $I_{c.o.m}=Ma^{2}({\frac {2}{5}}-{\frac {9}{64}})={\frac {83}{320}}Ma^{2}$. Now for a start, 83/320 seems like about the ugliest moment of inertia coefficient I've ever seen, so I'm pretty sure I've gone wrong somewhere here - but where?

ii) The hemisphere initially rests on a rough horizontal plane with its base vertical. It is then released from rest and rolls on the plane without slipping. Let $\theta$ be the angle that the base makes with the horizontal at time t - express the instantaneous speed of the centre of mass in terms of b and the rate of change of $\theta$, where b is the instantaneous distance from the centre of mass to the point of contact with the plane. Hence write down expressions for the potential and kinetic energy of the hemisphere and deduce that $({\frac {d\theta }{dt}})^{2}={\frac {15gcos\theta }{(28-15cos\theta )a}}$.

Now I really have no idea what on earth to do on this one, and I can see that part (i) is probably relevant but it's not rotating around the centre of mass ('C') so I can only assume they want us to calculate another moment of inertia which is just plain annoying. The picture I have has that the midpoint of the 'base edge' (where the centre of mass of a full sphere would be - call this point 'P', say) is always directly above the point of contact of the sphere and the surface - call this 'B' say - I'm pretty sure this is correct because at the very least it's true at the start and the end of the motion. In that case, by the cosine rule for the triangle CPB, the length of side CB squared is $b^{2}=a^{2}+{\frac {9a^{2}}{64}}-2(a)({\frac {3a}{8}})cos(\theta )=a^{2}({\frac {73-48cos\theta }{64}})$ which again seems wrong. Even if that is the correct expression for $b^{2}$, I'm not sure how to express the speed of the centre of mass in terms of $b$ and ${\dot {\theta }}$, and will the kinetic energy expression include an ${\frac {I{\dot {\theta }}^{2}}{2}}$ or an ${\frac {mv^{2}}{2}}$ or both? Apologies for my cluelessness, as I said this is a fair bit off the syllabus I'm meant to know! Thanks a lot,

Otherlobby17 (talk) 01:18, 4 April 2009 (UTC)Otherlobby17

Despite the moments of self-doubt the calculations done so far are all correct. The calculation of the speed of the center of mass is actually quite simple. The semi-sphere is rolling without slipping, which means that the contact point is momentarily at rest and the center of mass is momentarily in a circular motion around that point. $v=b{\dot {\theta }}$. Now we apply energy conservation. The amount of potential energy being released is $\Delta V=Mghcos\theta \,$, where $h={\frac {3}{8}}a$ is the distance between the center of mass and the flat surface. That energy released must be equal to the kinetic energy which is the sum of the rotational and translational kinetic energies $K={\frac {I{\dot {\theta }}^{2}}{2}}+{\frac {Mv^{2}}{2}}$. We have than

$2Mghcos\theta =Mb^{2}{\dot {\theta }}^{2}+I_{cm}{\dot {\theta }}^{2}=Ma^{2}{\frac {73-48cos\theta }{64}}{\dot {\theta }}^{2}+{\frac {83}{320}}Ma^{2}{\dot {\theta }}^{2}=Ma^{2}{\dot {\theta }}^{2}{\frac {448-240cos\theta }{320}}=Ma^{2}{\dot {\theta }}^{2}{\frac {28-15cos\theta }{20}}$

${\dot {\theta }}^{2}={\frac {40ghcos\theta }{a^{2}(28-15cos\theta )}}={\frac {40g({\frac {3}{8}}a)cos\theta }{a^{2}(28-15cos\theta )}}={\frac {15gcos\theta }{a(28-15cos\theta )}}$

Dauto (talk) 04:27, 11 April 2009 (UTC)