Shapes of Free Fall

good afternoon everybody Happy New Year and welcome to lecture four in this series on cosmic concepts the title of today's cosmic concept is shapes of free form well I want to begin by taking you to the elegant and geometric splendor of Wells Cathedral in Somerset in western England amid that elegant geometric splendor is a rather remarkable clock and this clock dates back some six centuries to when King Richard the second was on the throne this clock is remarkable not for its age not just for its age but also for its beauty and also for the fact that it tells us that the clock maker at the time thought that the Sun on the outer circle there orbited around the earth it was a time when the geocentric worldview held sway and it's understandable I think that the people of the time should have such a parochial view of the world given the technology that was not available to them at the time there's a similar sort of clock in Leo Cathedral which tells of a similar worldview the idea that the moon orbits around the earth fair enough but so too does the Sun orbit around the earth well it is possible to understand the validity of that worldview when people's experience was solely governed by seeing the Sun rise in the east in the morning and set in the West every evening it made sense at the time but our understanding has superseded that worldview now that we understand rather more about the contents of the solar system and it's the contents of the solar system and the orbits of bodies within the solar system that I'm going to be discussing in today's lecture so here is an image of the main players in our solar system the Sun is by far the most massive body in the solar system it's the only star in the solar system what else is in the solar system besides the star we call the Sun and besides the major planets well just to review our place in space the solar system consists of a somewhat ordinary star eight planets these days a number of minor planets a truckload of asteroids a huge number of comets and an uncountable number of rocky bits and pieces rocky detritus all of the things that I've mentioned can legitimately be regarded as being in orbit around the Sun but all of those bodies taken together don't account for barely one thousandth of the mass of the Sun itself by far the dominant mass in our solar system is the Sun itself the mass of the Sun is 2 times 10 to the 30 kilos 10 to the 30 meaning of one and thirty zeroes it's a huge number it's a huge mass everything else outside of it less than one thousandth of that and gravity from the Sun is the only acceleration mechanism that all the other bodies are experiencing unless you start to account through all those bodies and the different gravitational forces that one planet exerts on another those do exist but their vast least than the acceleration due to the Sun gravity is what calls the shots in terms of the motions of bodies within the solar system and so that's why I chose as the title for today's talk the idea of free fall and the shapes of free fall and I'd like to begin by just clarifying exactly what it is that physicists mean by free fall do we merely mean falling freely not so much in physics what we mean by free fall is to feel acceleration solely due to gravity and to nothing else the colloquial use of free fall is something a little bit different skydivers think in terms of flinging themselves out of a plane and falling to earth as free fall I have to say it's not my idea of freedom to throw myself out of a plane I would not regard that as falling freely I would be very conscious of the fact that gravity was drawing me to the earth even before the parachute is open in this scenario the person who tucks themselves out of a plane will still be subject to drag from air resistance as they fall towards the Earth accelerated under gravity of course when the parachute is open that drag is vastly and deliberately increased to slow down the net acceleration on the falling human so this is widely regarded as as freefall in colloquial terms but as far as the physicist is concerned we think of purely acceleration due to gravity as being what constitutes freefall and that of course encapsulate s-- objects that are in orbit where the only force that's acting is gravity itself and so we can think of the satellites in orbit above the earth as being in freefall this is a particularly well-known one this is an image of the Hubble Space Telescope which is in a low Earth orbit and every 90 or so 95 minutes it does a complete orbit all the way around the earth so in the time it takes for us to complete the question-and-answer session at the end of this lecture the Hubble Space Telescope will have performed about two-thirds of an orbit around the Earth if we go a little further afield beyond the low-earth orbit out to the kind of orbits where the so-called geostationary satellites are put those satellites that remain absolutely locked above a particular point in the equator they are much further out and so their orbital period is rather longer than an hour and a half it's much more like 24 hours that it takes for the earth to do a complete spin that of course comes from the definition of a geostationary satellite but if we go even further afield beyond the artificial satellites that we have launched from Earth and think about a natural satellite our only natural satellite the moon the orbital period the time taken to complete an entire orbit around the Earth is a lunar month just over 27 days if we plot the different distances of the three types of satellites that I've just referred to against the time to complete a whole orbit around the Earth and if I do that plot on logarithmic axes then I get an exact straight line the satellites that are much closer have a shorter orbital period the moon being much further out has a vastly longer orbital period this straight line relationship we now know Kepler's third law and Johannes Kepler is a key figure in what I'm going to discuss today but before we get to Kepler I want us to talk about Galileo people have been fascinated for centuries about how mass is pulled towards the earth under gravity about four centuries before I took this photograph of the Leaning Tower of Pisa in Italy it is said that Galileo dropped two spheres from the top and timed how long they took to travel down to the ground it said that these spheres were made of different materials and that they arrived at the same time demonstrating that if you only have acceleration under gravity objects of different masses will arrive at the same time it's interesting to note that this refuted Aristotle's notion that how the speed at which an object is pulled towards the earth is proportional to their mass this simple demonstration attributed to Galileo completely reviewed refuted Aristotle's theory but there weren't many bystanders at the time and arguably one could discuss the role played by air resistance in those two masses that Galileo dropped off this conveniently Leaning Tower a classier way to perform this experiment took place rather more recently not four centuries ago but four decades ago on the moon very conveniently for us in the context of performing this experiment there is no atmosphere on the moon and so this means that when the app astronaut David Scott took up and held in his right hand a hammer and in his left hand the feather of a falcon and then dropped the two simultaneously so that they could land so that one could see how quickly they landed on the surface of the moon knowing that there could be no air resistance because there was no air this experiment had a lot of validity the moon's gravity is much weaker than the earth and that's why it doesn't have any atmosphere and why there is no air resistance so this was a jolly neat way of demonstrating that acceleration due to gravity is independent of the mass of whatever it is that you're dropping if you open a parachute over that mass of course then things are very different but for this simple clean experiment this result became clear the acceleration is independent of the mass that you're dropping they're talking of the moon as we study the surface of the moon we can learn a lot about our surroundings in the solar system so let's start by taking a closer look at this so we perhaps can think that because we have the planets moving very sedately on their orbits that the solar system is very clean and tidy an ordered place not so much let's take a closer look let's zoom in on this beautiful image of the moon and sure enough when we zoom in we see that there are very many circular features on the moon known as craters there are vastly more craters on the surface of the moon than there are on the earth of course they're not unknown on the earth there's one in Arizona there's one in Mexico there's one in China but actually craters on the earth are very rare craters on the moon very common indeed why is this and how what's the reason for the difference in the number of craters on the moon versus craters on the earth well the key difference is that we have an atmosphere on earth this is jolly useful for breathing but it's very important for protecting us from the wilds of the solar system which are moving around outside of our consciousness to a reasonably large extent earth is sufficiently massive that its gravitational attraction is sufficiently strong that the atmosphere is retained why is this important in the context of avoiding craters happening on earth well let me show you something that I observed just about a week ago so a few days ago I returned from rural southern India where we had been working on a servicing mission at my school observatory in India that I described to you in my lecture just before Christmas my colleague Steve Lee and I commissioned a new all-sky camera a camera to look at the two pi/2 radians of sky overhead of the observatory so that we could always tell by night where the clouds were rocking in and set to jeopardise our observations watch closely on the left half of this image of the sky and you'll see that we we actually caught on the first night that we deployed this camera a meteor that meteor appeared this is a 40 second exposure which is one reason why the lights around the edge are so bright let me just zoom in on this meteor you can see that it's extent is about half of Orion which is lurking in the top right corner of this same image if i zoom in even further you can see why I'm confident it's a meteor because the brightness gradually increases from the top a little bit bumpy as we get more explosions right down to the brightest point at the bottom if it were a satellite or a plane you'd see very uniform brightness captured during the exposure of that camera so the reason why this was seen as a meteor was a little bit of rock probably the size of a grape thundered into Earth's atmosphere and because of frictional heating it got so hot that it burnt up in this spectacular fashion it's remarkable that just four nights later we saw yet another one so here goes watch this beautiful image you can see Orion in the center there and then gradually gradually going in the from bottom to top over on the left you can see another meteor again faint initially when it enters Earth's atmosphere and then brighter later on as the burning becomes even more ferocious if Earth's atmosphere had not caught up those bits of rock that formed these beautiful meteors which are spectacular to look at they could well have landed splat on the earth and made a baked great big crater possibly in someone's house or a school so it's very fortunate for us that we have an atmosphere which enables such rocky debris to burn up rather than hit us and do lots of damage just occasionally though meteors become meteorites and they hit the earth and they do a huge amount of damage so in February of 2013 there was a much larger spectacle and I'm showing here some stills from a dashcam that was captured by one of the many vehicles that witnessed this this scene although it's gone very very dark in this photograph II it's not because the earth went dark simply that there was so much brightness from this meteor that the auto exposure on the camera really tried to shut itself down and take a very very tiny exposure this particular rocky entity that entered Earth's atmosphere is thought to have been 20 meters in diameter on entry to our atmosphere it was 30 times brighter than the Sun when this photograph was taken it caused a lot of damage in terms of radiation burns on people's skin and on their retinas three-quarters of this 20 meter rock were vaporized because of frictional heating due to Earth's atmosphere thankfully no one was killed so rocks will get attracted any kind of massive object will get attracted towards the Sun at the center of our solar system but if a planet such as Earth such as Jupiter gets in the way or such as Earth's moon gets in the way and a big splat will happen as that rock collides with that planet or with that satellite what else gets attracted towards the Sun well something else that is a constituent of our solar system are the comments I'm going to show you now a time-lapse of some images taken by my colleague Steve Lee of a comet that he discovered in 1999 these are lots of separate stills which show the passage of the comet moving against a fixed star background he spotted it entirely serendipitously he knew immediately it was a comet because he knows this patch of sky very very well indeed and we do know that fortune favors the prepared mind he spotted it a long way out a long time before it orbited around the Sun and so that meant it could be studied for a long time before and after going around the Sun this comet is very very poisonous contains a lot of different cyanides it's about two kilometers across so it's probably just as well for us it didn't come anywhere near the earth at two kilometers it would not have burned up completely as a meteor it would have ended up as a meteorite if it had in fact impinged on earth this comet is now heading away from the solar system and it's recently gone past Saturn comets can be really very spectacular this is another one that was discovered in 2011 by Terry Lovejoy photographed again by Steve Lee from the anglo-australian telescope in Australia this one is thought to orbit around the Sun with a period of just over six centuries so when it was last rocking up near Earth that was around the time that that geocentric clock in Wells Cathedral was being admired for the first time comets have long been regarded with or and sometimes fear by humans on this planet nineteen years before Gresham College formed in 1577 there was a spectacular comet and one of the early witnesses of this comet was six year old Johannes Kepler and surely this big comic depicted here made a very big impression on him given the later input and energy and dedication he put into studying the orbits of bodies through the solar system Johannes Kepler was a German mathematician and astronomer and he worked closely with the Danish astronomer Tycho Brahe Hey first as his assistant but later after bra haze death doing lots of calculations on data collected by bra hey bra he'd had a passion for accurate measurements and observations he painstakingly measured the positions of planets and of stars that were remarkable for their quantity and for their quality showing remarkable precision given the era at which he conducted these measurements this is the observatory on the island of hven at the time in Denmark now part of Sweden where all these wonderful observations were taken that was so important for johannes kepler's later work Tycho Brahe he was an extremely accurate observer he was truly observant and he made copious careful notes on the left here is a copy of some notes he made about that comet that made an impression on the six-year-old Johannes Kepler he noted that the tails of comets always point away from the Sun so the tails of comets are not subject to gravity primarily but they wish away from the Sun because of the solar wind that blows them away and because of radiation pressure from the photons that are streaming away from the Sun so that's why comet tails always appear to point away from the Sun and Tycho Brahe hey figured this out quite a few centuries ago but back to Kepler as I said he made extremely exquisitely careful calculations which made a big leap over the current understanding of the Copernican idea of orbits as being purely circular with the Sun at the center turns out there is more subtlety and more depth and more richness to the paths of orbits the paths of free fall within the solar system that we'll come to in just a few minutes so the big insight that Kepler was able to reach was the fact that the orbits of the main planets around the Sun weren't straightforwardly circles they were much more accurately described as ellipses with the Sun not at the center of an ellipse but actually at a focus of the ellipse and I'll either illustrate little later on this in this talk what I mean by the focus of her lips but before we get to that I want to introduce another historical figure this time the Greek mathematician Apollonius of Perga now Apollonius of Perga was a mathematician he was a geometry he loved geometry he worked out some very important mathematics many many centuries the best part of two millennia before they became relevant for describing the physical universe and this is actually pretty common in the history of mathematics that subjects that are first explored for their own intrinsic beauty are later found out to be exactly the ones needed to describe this amazing physical universe in which we live and the particular branch of mathematics which is relevant to our lecture today that Apollonius worked on is the theory of the so-called conic sections conic is simply the adjective corresponding to the noun cone a cone is made by sweeping out a line with respect to an axis of symmetry we'll see that a little more in a second the sections the conic sections are the sections that you get when you slice a cone with a plane so let me show you first of all very simply what happens if you take a cone and then you pass horizontally with respect to the symmetry axis of that cone what shape you get I hope you get a sense with the perspective of this drawing here of the fact that the shape that the the red edge of the yellow circle makes with that cone is a familiar circle let me show you hopefully a little bit more graphically exactly what this shape is so I brought along with me today a conical flask filled with water and pink food coloring if I hold this exactly horizontally you will well recognize that the the shape delineated here is that of a circle supposing I then hold the cone at an angle you can I hope then see or at least imagine that the shape where the liquid intersects the cone of this conical flask is no longer circle but something very elongated still with two axes of mirror symmetry but it's now what we call an ellipse the fun doesn't end there let me now move to other types of symmetry so we're now using green food coloring my father and I spent a certain amount of time over the Christmas holidays thinking about just how much fluid we needed in one of these conical shapes if I line up the surface of this with the surface edge of the cone this surface age is known as a generator because you generated the cone by moving a line about this axis of symmetry then the shape that you get by the intersection of the fluid with the shape of this flask is a parabola if I do an even more extreme shape then the shape that I get is a hyperbola now why am i waving these conical flasks of rather intently colored fluid in front of you it's for an important reason the shapes that are represented by the surface of the fluid as I hold it at different angles are the shapes with which bodies move through the solar system so if it's possible to return to the main slides then you get a sense this is an image of a hyperbola a section through a cone the word hyperbola comes from the Greek meaning exceeding and what you'll notice about this hyperbola is that it isn't a closed orbit like the circles obviously were and like the ellipses obviously were but it's an it's a it's a line that just goes straight on straight on for however long the cone is turns out that hyperbola is the shape of the path that our great many comets actually exhibit as they move under the gravitational attraction of the Sun through the solar system apart from the ones that are ellipses that is so I just want to take a moment to lead you through some of the incredibly beautiful mathematics that Apollonius explored best part of two thousand more than two thousand years ago please don't worry if equations aren't your thing there is only one equation and it's a very beautiful one so I want you to think of are on the left simply a distance think of it as a radius that radius may change a little bit as we move that lilac line through different angles theta we normally represent an angle theta by a Greek by a Greek letter such as theta I want you to think of K as just a constant you can set it to the number one if you like all it all it does is to scale the overall size of a shape but it's a constant the letter E is important that's going to be fixed for a particular kind of shape so all that's really going on here is that the angle theta cycles through as we trace out 360 degrees with respect to the horizontal axis on the right-hand side of this slide if you don't know what the word causes again don't don't worry it is simply something that is telling us about about how the value of this function changes as you cycle through the angles it takes the value 1 when theta is zero it takes the value minus 1 when theta is a hundred and eighty degrees so it cycles through different values now by far the simplest of the conic sections that I've told you about is the circle and that happens very simply when e is 0 when e is 0 the entire bottom denominator of this fraction goes to 1 and that simply means the radius is a constant and we know that well for a circle if ever we've spun a conch around a finite bit of string above our heads we're used to the fact that a circle has a completely constant radius as we trace out its path so the circle is by far the simplest of the conic sections what about increasing the difficulty just a little bit more let me take the same equation exactly the same equation we had before but now instead of setting the value e to 0 we're going to set it to some number between 0 and 1 take 1/2 if you like so that means that we're going to get a different shape according to the value of this angular term on the bottom and we're going to get something that's rather elongated when theta is near 0 or where theta is near 180 degrees specifically we're going to start tracing out this beautiful shape of an ellipse this is the shape with which a good many planets and a good many moons of planets orbit around their central mass the equations I've been showing you for conic sections I chose because it's a simple one that conveniently represents all the different conic sections but there's an entirely different equation of an ellipse that takes into account what are called the Cartesian coordinates the coordinate system of rainy day car when you think about X along a horizontal axis and Y along a vertical axis and the important thing about that representation is that you can very clearly see how rather than having a single radius that defines the circle we now have the two lengths of this red line and it's their total length that is a constant and thereby defines the path of the ellipse the path of the orbit that elliptically orbiting bodies follow so this is widely general to the paths of bodies orbiting in the solar system around the Sun it's true for a number of the moons around Jupiter I showed you this image in my very first lecture these bright make four moons here are the so-called Galilean moons and their orbits are fairly circular but some of the other Jovian moons further out are very much elliptical orbits very highly elliptical indeed their evaluate a have high eccentricity and these orbits have generality not just within the solar system but throughout our galaxy and beyond but I just want to take you back to a slide I showed you in my lecture just before Christmas talking about black holes and particularly showing you these time-lapse images of the stars very close to the black hole at the centre of the Milky Way I showed you this image which takes together all those successive images of the positions of stars at the very center of our galaxy and you can see that there are some really quite elongated ellipses ellipses with very high eccentricity but in the real physical universe our tracing out these elongated paths they're not circles the black hole is not at the center the black hole is at the focus of one of the two foci of the overall elliptical paths that are drawn there so if this is what we see for the solar system for the for the main bodies in the solar system the planets themselves which are predominantly elliptical what of what of a situation where we have say just two celestial bodies two stars orbiting one another or perhaps just the earth-moon system I'm going to show you a movie now which has two bodies of different mass shown from the perspective of their center of mass and you can so the center of mass also called the barycenter is illustrated by this blue cross the green body is the more massive of the two bodies and the red one is the less massive of the two bodies but if you just stare at any one of those bodies you'll see the paths that they trace out are ellipses we can do a similar thing by just imagining that we keep the green body constant and again you can see the red body is orbiting around it it's not orbiting the center of the elliptical path that it traces out it's orbiting the green object which is at one of the two foci of the ellipse in a difference so you can imagine that as being a heliocentric view if you like now let's look at similar kind of movie but now from the vantage point of the red body if you like this could be a geocentric view there's validity in considering the motion from all different perspectives obviously bearing in mind that the underlying mass responsible for really calling the shots in these orbits will be that of the more massive body and so according to if you have two celestial bodies coming together and orbiting one another the exact shape of their orbits the exact shape of their free fall will determine very much be determined very much by the initial energies of the two bodies when they came together it'll be determined by the momentum or more specifically the angular momentum the speeds with which the bodies have with respect to one another and that's what determines whether we have a very round orbit with a low value of e a low eccentricity or a very extreme ellipse with a value of e that's much much closer to 1 or indeed something rather more spectacular an orbit that is unbound which is the case we're now going to consider back to this simple equation if I takes a value much much larger than 1 this orbit will never close it will never be bound it will never permit a celestial body a celestial body that's following such a path will never retrace its steps in a closed orbit and this is the case for a hyperbolic orbit which is what that great comet of 1577 was calculated to have been following and indeed still following using the accurate data collected by Tycho Brahe hey that I mentioned earlier modern day scientists at JPL in California were able to calculate what they think that comet trajectory that comet path actually is and so it's now thought that this comet is over 300 astronomical units away from the Sun if you're not familiar with what an astronomical unit is let me just tell you that one astronomical unit is basically the separation of the Earth from the Sun so 300 of those is a very long way indeed way beyond Neptune long long way away so let's just summarize the conic section that Apollonius gave to us as a mathematical gift couple of millennia before he could have known that actually these mathematical descriptions have the most amazing validity in the context of a description of the real physical universe this one equation describes all of the conic sections the circles the ellipses the parabolas the hyperbolas how'd you get to all of those from this one equation you have four different families of values of e e as I said is the quantity known as the eccentricity of the geometric shape being traced out K is a constant remember that just scales the size and so this polar equation this representation of these shapes solely relies on the angle theta just tracing out all the values from naught to 360 but being scaled according to its value by whatever the value of e is set to a simple case is when the eccentricity is 0 that gives us a circle between naught and 1 we have an ellipse sometimes a very extreme ellipse if the value of e is very near 1 if you could fine tune the value to be exactly 1 then you'd have a parabola but much more commonly for the Comets and the asteroids that orbit towards the Sun is a value rather than rather greater than 1 tracing out the shape of a hyperbola well it's time to introduce one more historical figure and that's the figure of Robert Hooke Robert Hooke was a pallette polymath in Oxford quite a few centuries ago and 350 years ago when he was giving a Gresham lecture he explained that gravitation was something that applied to all celestial bodies all massive bodies in the sky but he added to that the fact that gravity the strength of gravitational attraction or the gravitational acceleration would decrease with increasing distance he thought very deeply about these matters and by 1679 he thought that gravity actually had what physicists now call an inverse square dependence on the separation of the two galaxies what do I mean by an inverse-square law what I mean is that if you have the distance of say two stars that are orbiting around another if you could just do that in your imagination if you have that distance then you will increase by a factor of four the gravitational attraction that they feel towards one another that's all that we mean by an inverse-square law well Robert Hooke communicated this in a letter to the Cambridge polymath Isaac Newton and Isaac Newton thought very very deeply about what Robert Hooke had told him he thought very deeply about this business of acceleration depending on the reciprocal of the square of the distance that was separating two bodies a plot of this equation appear is shown in the bottom left corner of my slide on the basis of pondering these things very deeply Newton developed a whole new branch of mathematics which we now call differential calculus and Newton was able to show by the inverse square law of gravity that I've been referring to that the solutions to these equations must necessarily be the conic sections worked out by Apollonius couple of thousand years previously so the mathematics was in place the astronomical thanks to Apollonius the astronomical measurements were in place thanks to Tycho Brahe Hey and a succession of intellectual contribution from Kepler from hook and from Newton meant that it was now clear why the orbits of bodies in the solar system were the shapes that they are circles and ellipses for planets and for their own satellites for comets which typically emerge from outside of of Neptune the paths that they follow are either quite extreme ellipses or if they're eval use are higher than 1 then hyperbolas never ever to return to orbit around the Sun ever again well I want to finish by telling you very briefly about something that my students and I have been working on in Oxford and that's the business of what happens if you're orbiting around more than one Sun so supposing you have two stars orbiting around one another in a normal binary star system and of course they'll be obeying the elliptical paths described by the equations and the movies that I've shown you previously but suppose that you've got some little pebbles or rocky debris or gas that's in orbit outside of that in a binary of stars I'm going to show you a movie now put together by one of my students or gustas Porter and before I set it going I just want to explain what we're looking at so to start with just look at the left panel and the two white circles that overlap one another just for simplicity I've drawn their paths as circles and depending on the the set up initially it could entirely be that you have two stars in circular orbits around one another but the Green Line is going to trace the pattern of gas in orbit around that inner binary of stars the panel on the left is as it were a bird's eye view looking from overhead the panel on the right is edge-on view where you're seeing the stars much more like that instead of like that in each case that all the lines are the same but viewed from a different perspective watch what happens to the green line the orbit the path of free fall for matter outside of that in a binary system of two stars so here we go looks like the at an instant the green shape is fairly elliptical but it moves and in fact what you see is a precession so the shape of orbits outside two suns or two stars no longer follows a simple conic section confined to one plane but the plane itself is processing in and out of the plane contained by the binary we have discovered examples of this within different stellar systems in the galaxy they're awfully good good fun to study because you can consider the dynamical influence of the externally orbiting material following the green path on the inner binary itself and it's not just a dynamic a dynamical influence when stars get close to these circumbinary orbiting masses then there's a bit of an interplay in terms of their radiation well I do hope that I've been able to show you today that freefall is not merely a straight line like jumping out of a plane or dropping a mass from a leaning tower somewhere in northern Italy Isaac Newton demonstrated indeed stressed that the all the curves of the conic sections are possible paths of free-for-all the conic sections are possible orbits and as we study more and more objects within the solar system and beyond the solar system we see that that is absolutely true increasingly so but now we understand that the shapes of free fall and be even more complicated curves if you have something even more complicated at the center than just our see our nearest star the Sun so I do hope you've enjoyed learning a little bit about the shapes of free form thank you [Applause] [Music]

Channel: Gresham College

Views: 10,030

Keywords: gresham, gresham college, education, lecture, public, london, debate, academia, knowledge, Astronomy, Science, Physics, Katherine Blundell, orbits, planets, space, stars, Isaac Newton, energy

Id: u-JT62f6oKk

Length: 49min 10sec (2950 seconds)

Published: Wed Jan 29 2020