Why you weigh less when travelling east

what do hurricane winds and your weight have in common in this video we'll find out thanks to brilliant dork for sponsoring this video if you've taken physics to high school level you'll know there is a difference between mass and weight mass is measured in kilos or pounds if you're from the 19th century while weight is measured in Newtons now we confuse those two all the time because in everyday speech when we ask a question like how much do you weigh we often give an answer like how bloody dare you instead of something like 800 Newtons we do this because a person's weight and a person's mass are in the same direct proportion everywhere on earth weight is a force which means to calculate it we can use Newton's second law F equals MA where the force your weight equals your mass in kilos multiplied by the force of the Earth's gravity which isn't actually a force it's an acceleration man we are really about according things by the right name in case that's confusing think about it like this if you were to jump out of a plane and ignore air resistance then you would start accelerating and falling faster and faster at the rate of the Earth's gravity which if you're an engineer is ten meters per second squared so after one second you'd be falling at ten meters per second after two seconds you'd be falling at 20 meters per second and so on that means that to calculate the person's weight we can just multiply their mass in kilos by ten meters per second squared the acceleration due to gravity to get their weight so if you have a mass of 80 kilos that means you have a weight of 800 Newtons but if the acceleration due to gravity is the same everywhere it just makes sense to keep the numbers small to talk about kilos instead of Newtons fortunately however not all of us are engineers some of us are physicists that's worse that means we know that the acceleration due to the earth gravity isn't actually 10 meters per second squared it's closer to 9.81 but also that it isn't the same everywhere in fact the closer you get to the equator the Earth's gravity gets weaker so even though you would have a constant mass as you get closer to the equator your weight would decrease because the acceleration that's acting on has decreased this happens because the earth isn't stationary it's spinning on its axis and just like the circular motion of a merry-go-round the rotation of the earth provides a centrifugal acceleration which increases the further out from the axis of rotation you are the acceleration is given by the formula V squared over R where V is the velocity and R is the radius of motion it acts to try and fling you off the earth but because the earth isn't spinning very quickly it's not a very big acceleration if you near the Earth's poles then you're not far from the axis of rotation which means that you don't experience much of this centrifugal effect well if you're near the equator you experience a lot of it and because this acceleration acts outwards away from the earth while the force of gravity acts inwards towards the earth that means your mass is acted on by a net smaller acceleration and so you weigh less how much less well Earth's surface gravity at sea level will varies around nine point seven eight to around nine point eight three meters per second squared it also varies a bit without a tooth so we're talking a variation of about half a percent and that has actually been compounded by the fact that the Earth's rotation causes it to bulge at the equator the Earth's not a sphere is technically an oblate spheroid what that means is that at the equator there's a greater distance between you and the Earth's center of mass then there is at the poles and so because gravity is acting over a greater distance it means that the gravitational acceleration is a weaker but it gets stranger still because as well as weighing less the closer you get to the equator you weigh less the faster you travel east this was discovered at the turn of the 20th century when a series of research ships from the Institute of geodesy in Potsdam which I can only assume is one hell of a party school work performing experiments across the world's oceans on the strength the Earth's gravity back home analyzing their data was a Hungarian physicist named Baron Rowland on eat fish and he noticed that as well as the Earth's gravity being weaker the closer that the ships got the equator but gravity measured was weaker when they were traveling east and stronger when they were traveling west consistently across all their measurements you can actually see this really clearly in this one graph from a repeat experiment in 1908 originally the science ship is moving slowly westward then quickly westward then reverses direction and moves slowly eastward over all the ship experiences a change in gravity of around a hundredth of a percent its Bush cleverly identified that this effect was also because of the Earth's rotation but in a more subtle way when we're thinking about motion on the Earth's surface such as how gravity works we have to remember that our coordinates are fixed relative to the Earth's surface so latitude longitude height but that the earth is also rotating what that means is that we're working in a non inertial reference frame and it's called that because we are experiencing on the surface and acceleration when seen in a reference frame that's exterior to the earth such as the reference frame of the solar system what that means is that we have to make some corrections to the equations of motion that we use to for example calculate local gravity what I want to do is take you through this derivation because it's really quite simple but I think really quite cool and at the end of it there's a connection between two bits of physics that you wouldn't expect let's say that you're on the surface of the earth and have some north-south velocity lowercase V and an east-west velocity u that would mean that you're engaging in circular motion around two different axes your north-south velocity V is effectively moving you in a circle with a radius the same as the Earth's which we can call our your east-west velocity U however is taking you in a circle whose radius depends on your latitude it's radius is R times the cosine of your latitude when you're at the equator latitude zero its radius is going to be R while if you're at the pole latitude 90 its radius is zero that means then that by moving along these two circles you're also creating some new centrifugal acceleration radially outwards in the case of your north-south velocity and along the east-west axis but still outwards in the case of your east-west velocity except your velocity in the east-west direction isn't only your own because remember the plier is rotating so while your centrifugal acceleration radially outwards is given by the normal formula that's V squared over R your centrifugal acceleration in the east-west plane is given by a more complicated formula it's your total the velocity squared over the radius of motion so that means you plus the velocity you have because of the planets rotation all squared divided by R times the cosine of latitude well write U sub P for the planetary velocity for now remember though we're interested in the extra centrifugal acceleration that you experience because you're moving so we need to subtract the centrifugal acceleration that a stationary object would experience anyway just from the rotation of the earth but that's easy to do we can use the same formula and just set the local velocity U to zero expanding out this bracket then two of these terms cancel so we're left with just the one square term and this cross term so these are the two extra centrifugal acceleration you create by moving around on the surface of the rotating earth but we're interested in the effect this has on your weight which means the effect that they have on the acceleration you experience radially as that's the direction gravity acts in towards the center of the earth to calculate this is a simple bit of geometry we already have one component of acceleration acting radially V squared over R and one which is inclined at an angle to it a little bit of trigonometry shows that this angle is none other than your latitude and so the final change in gravity experienced because of your velocity is given by this formula where we've multiplied the second acceleration by the cosine of latitude we can make this look a lot nicer by substituting in the equation for u sub P which comes from circular motion around your latitude and writing your total velocity as calculated by Pythagoras as capital V we're left with this nifty little expression which has two components the second component is basically the centrifugal acceleration you need to create in order to remain on the Earth's curved surface well the first term is more interesting the first term is effectively the Coriolis effect just as poleward motion is horizontally deflected by the earth's rotation east-west motion is vertically deflected so your local gravity is deflected in much the same way that the winds that form a hurricane are and compare this expression to the Coriolis parameter which we use in meteorology for that purpose though not by as much if you were to travel due east at 20 meters per second which is 72 km/h 15 or which is London then you would experience a decrease in local gravity of naught point naught naught to 9 meters per second squared or equivalently if you have a mass of 80 kilos you would weigh naught point to 3 Newton's less which is still really cool so perhaps the takeaway from this is that if you want to lose weight then people are right to advise that you go running just to maximize your results make sure your head east fast if you enjoyed the derivation in this video then you almost certainly liked learning about cool bits of maths and physics through worked examples and if that's you then ho boy do I have an announcement for you introducing brilliant dog that's right it's a whole website of maths and science problems they're interactive they're beautifully illustrated they're expertly written I can't keep up that intensity the derivation we did in this video involved both gravitational physics and also rotating reference frames in classical mechanics and when you know it brilliant has expertly written modules on both of these topics so if you'd like to explore more about how the earth rotating causes some unusual physics then you can learn through solving problems and testing yourself it's honestly the single best way to learn having said that I've long wanted to brush up my iron oxide II chemistry so I will be checking out their new chemistry course if you'd like to join me then held on over to Berlinda org slash Simon Clark and the first 200 subscribers to do so will get 20% off their premium annual subscription or if you want you could buy a subscription for a friend or a loved one perhaps even an engineer so they could learn some proper science thank you for watching this video I made it because of a chance discovery I made on reddit so that took me down a neat little rabbit hole of planetary dynamics which I hope you found interesting too if you did find it interesting if you learn something then pop it alike give it a share to people you might find interesting and you can always subscribe to this channel if you want to see more of this kind of thing I guess that just leaves me to say thank you again for watching and I'll see you in the next one

Channel: Simon Clark

Views: 33,712

Keywords: dr simon clark, drsimonclark, simonoxfphys, simonoxphys, video essay, eotvos effect, Eötvös effect, planetary dynamics, physics, science, coriolis force, coriolis effect, coriolis, rotational dynamics, centrifugal, centripetal, xkcd

Id: ok7ONqtVvEo

Length: 11min 46sec (706 seconds)

Published: Tue Feb 18 2020