Lecture 17 Rotational motion Kepler s Laws Some history Study of the motion of the planets has a special place in history due to its impact on science and as an example of conflicts that can arise between science and religion Astronomy has been important since the dawn of civilization and quantitative scientific studies of the motion of the planets occured in many early civilizations for example in the Chinese Arabian and Incan cultures Ptolemy 85 160AD put forward an earth centered geocentric model of the motion of the planets As measurements of planetary motion became more precise it was clear that Ptolemy s model was wrong The Polish astronomer Copernicus 1473 1543 put forward a theory based on the idea that planets moved around the sun in circular orbits heliocentric theory The Catholic Church had adopted the Ptolemy model as an article of faith as had most other religions and scientists of the time so Copernicus was considered a heretic Tico Bahe made meticulus measurements which supported the heliocentric model of Copernicus which was greatly elaborated upon and supported by Galileo using his new telescope Galileo was brought before the Italian inquisition for his belief in the heliocentric model Kepler showed that circular orbits do not explain the observations and instead proposed his famous three laws which are now known to be accurate Kepler was Lutheran and German and also suffered considerably because of his belief in the heliocentric theory The study of the motion of the planets provides a important example of the context within which science works with conflicting theories interpersonal conflicts sometimes including conficts with religion and with the broader society Another truth is old theories never really die and by surfing the internet you can still find sites vacuously articulating opposition to the heliocentric model and supporting the view held by the majority of scientists and religious figures of the sixteenth century Another amusing topic is the flat earth theory which was actually rejected by most intellectuals of the later Greek and Roman civilizations It was somewhat popular in the very early middle ages but was not popular at the time of Columbus though many people incorrectly believe that it was religious dogma at the time Of course evolution is a much more topical conflict between science and religion in our society 1 Kepler s laws i All planets move in elliptical orbits with the sun as one of the foci ii A line drawn to the sun from a planet sweeps out equal areas in equal times iii T 2 r 3 where T is orbital period and r is the average distance from the planet to the sun Kepler s laws apply to a small mass e g a planet moving around a large mass e g the sun They apply to moons moving around planets asteroids moving around the sun etc Here we will consider the special case of circular orbits and demonstrate that if i is true the ii and iii follow In later lectures we shall return to the case of elliptical orbits Consider a planet moving in a circular orbit of radius r with constant speed v as seen from the sun If the orbital speed is constant and the motion is circular then no proof of ii is needed Proof of Kepler s third law follows from Newton s second law and the formula for centripetal acceleration GM m mv 2 r r2 1 We find the relation to the period T by using velocity distance time so that 2 r v 2 T Using this to remove v and cancelling the mass m we find that T2 4 2 3 r Kr 3 GM 3 For the case of the sun the constant K becomes Ks 4 2 GMs 2 97 10 19 s2 m3 It is possible to find the mass of the sun by knowing T and r both of which can be deduced from observations of planetary motion Furthermore similar methods can be used to deduce the mass of any planet which has a moon The period of the moon can be measured and the radius of the orbit can be measured when the moon is adjacent the planet A couple of problems 2 Problem Held up in a rotating cannister Given a static friction coefficient of s 0 5 find the angular velocity required to prevent a person slipping down the side of a cylindrical cannister of radius r 5m when the cannister is vertical Solution The acceleration required to produce circular motion is v 2 r The associated force is mv 2 r The normal force of the person on the cannister is N mv 2 r so the friction force is s N s mv 2 r The force of gravity on the person is mg so the rotational speed required to resist the downward force of gravity through friction is given by s m v2 s m 2 r mg r 4 gr 1 2 9 9m s s 5 so that v Which implies that v r 1 95rad s 18 6rev min At fixed angular velocity the larger the radius the greater the force Problem Where the rubber meets the road i Cornering depends on centripetal acceleration and on friction There are many variants of these problems First consider using only friction to corner What friction force is supplied by the tyres on the road This is easy enough fs mv 2 r 6 We might also be asked what value of s is required in which case we can use fs s mg mv 2 r so that s v2 rg 7 Note that this is independent of mass so that heavy cars and light cars need the same quality tyres at least when cornering Other interesting issues to do with friction are related to braking For example the coefficient of static friction between a good dry asphalt road and tyres is about s 0 8 while on a wet road it is a half that or smaller Also since the static friction coefficient is higher than the kinetic friction coefficient it is better to brake without skidding 3 ii A second variant of these problems is to reduce the friction force required of the tyres by banking the turns For example in NASCAR Daytona where the average qualifying speed is about 200mph the tightest turn is banked at about 31 degrees At michigan International speedway average speed about 180mph the tightest turn is banked at about 18 degrees Highspeed trains also have banked turns and of course roller coasters have the ultimate in banked turns including upside down A simple case is to ask In a spiral roller coaster with radius r 4m which is the minimum angular velocity required to prevent the roller coaster from falling off the track At the top of the circle there is an acceleration g downward This acceleration can produce circular motion or it can cause the coaster to fall In order for it to produce circular motion we need to be moving fast enough ie v2 g 8 r The smallest velocity is then vmin 9 81 4 1 …
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