ShmoopTube

00:02

Newton's universal law of gravitation man Newton created more

00:07

laws than Congress...[mumbling]

00:37

you ever get tired of buttering toast yeah me too [Man with buttered toast in a kitchen]

00:41

every morning taking that cold hard butter from the fridge and trying to

00:44

spread it on a nice piece of sourdough and only there was an easier way but now

00:49

there is thanks to the revolutionary new invention the butter gun 3000, top notch [Man holding a butter gun 3000]

00:54

German engineers have built the butter gun using an inverse square law all

00:58

right well what is an inverse square law well let me show you say you've got one

01:02

piece of toast and one stick of butter well you just load that stick of butter

01:06

in the butter gun 3000 stand one meter away take aim and shoot some people love [Man shoots butter gun at slice of toast]

01:11

a whole stick of butter on their toast and those people probably have heart

01:15

disease just keeping it real but if we want to spread that butter out well we

01:19

can double the distance from the toast now we just nail four pieces of toast to [Woman holding board with 4 pieces of toast nailed]

01:23

our handcrafted breadboard that chips free with every order and we take our

01:28

shot see when the distance is doubled were

01:31

able to quadruple the amount of delicious butter breakfast good..Of

01:36

course each piece gets 1/4 of the butter but you know say you've got a big hungry

01:41

family well four pieces of toast just won't cut it just take one more step [Man aims at bread board with butter gun]

01:46

back and triple the distance and then you're able to butter nine pieces of

01:50

toast at the same time and each piece gets 1/9 of the butter

01:54

well this relationship between the distance from the bread and the amount

01:57

of buttered toast works just like gravity both follow an inverse square

02:02

law so let's look at that butter gun again

02:06

at first when I was at a certain distance away which we'll call D...One

02:11

piece of toast got all the butter and I buttered the heck out of it but [Man holding butter gun in kitchen]

02:15

when the distance was doubled each piece got 25% of the butter and at triple the

02:21

distance the pieces of toast got 1/9 of the butter each so there's pattern here

02:26

as distance increases each piece of toast gets an amount of butter that

02:30

equals the inverse of the square of the distance well this is an example of an

02:35

inverse square law and there are lots of inverse square laws that don't involve [Man discussing inverse square laws]

02:39

butter at all the amount of light we get from the Sun or any light source works

02:43

as an inverse square law same thing with electrical force and the most famous

02:48

inverse square law relates to gravity well let's say we have two planets that are [Two planets spinning]

02:52

exerting gravity all over each other then we double their distance the force

02:57

of gravity will be 1/4 as strong triple it? yeah 1/9 quadruple it? well the

03:03

force of gravity would now be 1/16 of what it was and the info-mercial

03:07

business sure doesn't work like that we'll throw in one free butter gun if [Man advertising free butter gun]

03:12

you order now but you'll still have to pay for shipping and handling we're not

03:15

reducing the price of anything to 1/16 no matter how far away you are well

03:20

Isaac Newton took this concept and came up with a whole equation for the force [Isaac Newton appears in field and apple hits his head]

03:24

of gravity between two masses Newton's universal law of gravitation says that

03:29

the force of gravity between any two objects equals the gravitational

03:32

constant times the mass of the first object and the mass of the second object

03:38

and all of that is divided by the square of distance between the centers of each

03:44

object and this equation applies to any two objects anywhere in the universe so [Newton's equation appears]

03:49

right now at the center of the Milky Way 26,000 light-years away there's a huge

03:55

black hole called Sagittarius A and that huge black hole is exerting a

04:01

gravitational force on you yes you and you know what else you're exerting a [Sagittarius A exerting force on woman]

04:07

very small gravitational force right back to it

04:10

yeah just sitting there your body is pulling on a black hole that's billions

04:14

and billions and billions of kilometers away sure it's a teeny tiny pull that

04:18

your body is exerting but still the universe is crazy y'all well it almost

04:23

makes this whole butter gun invention seem insignificant [Man looking sad with butter gun]

04:27

and because of that insignificance if you call in the next 30 seconds we'll

04:31

take $10 off price okay stop with that but you might have noticed that we kind

04:36

of just breezed past something just now the gravitational constant also known as

04:41

the big G to differentiate it from the little G which is the acceleration of

04:45

gravity for objects on earth...Well, the gravitational constant is 6.67408

04:51

times 10 to the negative 11th cubic meters over [Gravitational constant appears]

04:57

kilograms and seconds squared it can also be written like this with meters

05:02

cubed kilograms to the negative first and seconds to the negative second

05:06

well those negative exponents show that these units are inverses so that's kind

05:11

of a weird bunch of units to you know just slap together there but they're

05:15

there for an important reason the universal law of gravitation an equation [Law of gravitation formula appears]

05:19

finds the force of gravity between two objects and forces measured in Newtons

05:25

and the Newton is a derived unit made up of kilograms times meters over seconds

05:31

squared well if we take big G out of the universal gravitational equation and [Big G disappears from equation]

05:36

just look at the units of what's left we find two masses that will be measured

05:41

in kilograms and radius squared which will be measured in meters well since

05:46

the masses will be multiplied we can consider that to be kilograms squared we

05:52

have to get from kilogram squared over meter squared to kilograms times meters

05:56

over seconds squared so our force measurement uses the right units..Well, we

06:01

do that by giving the gravitational constant the necessary units to get us

06:04

there and by doing unit analysis we can see that when we add the units in the

06:09

gravitational constant back to our universal gravitation equation [Units added to gravitational equation]

06:13

well we end up with the unit's needed for Newtons these units were assigned to

06:18

the gravitational constant so that we end up with the right unit for force as

06:22

far as that value the 6.67408 yeah well that's been found by doing very

06:28

careful measurements I can't go into the history of all that measurement because

06:32

changed a bit overtime and it stirred up a little controversy here and there but [Men wearing wigs sit down at a table]

06:35

if you're curious fire up the machine and read up on the wonderful world of

06:39

the gravitational constant and you probably should find a hobby or

06:43

something else to do with your life too all right now for our purposes we're

06:47

just gonna round this down to 6.67 times 10 to the negative 11

06:51

and we're not going to list the unit's every time trust us they're not going [Man walks to computer chair]

06:55

anywhere so let's take a look at how this universal law really works well

06:59

let's say we have two planets that are distance D apart if we double the

07:03

distance between the planets and triple one planet's mass how does that change

07:06

the force of gravity between them hmm we don't really have a lot of numbers [Man walks up to window]

07:10

here to work with yes this one is one of those big picture concept type of

07:14

questions but we know the gravitational constant 6.67 times 10 to

07:18

the negative 11 blah blah blah but even that's not going to be useful here

07:22

so let's make this real simple for ourselves we're just gonna make

07:26

everything equal to one even big G well let's use the universal gravitation

07:31

equation the force of gravity equals big G times mass one times mass two and all [Gravitation equation appears]

07:38

that's divided by the distance squared well when we turn all those letters into

07:43

ones we find a force of one Newton now let's make the changes well change the

07:48

first planet from one to three and we'll change the distance from one to two and

07:52

of course we have to square that distance now when we do the math we find

07:56

the new force is three over four Newton's aka 3/4 so even though we're

08:01

tripling one mass and only doubling the distance because of that inverse square

08:05

law it results in a reduction of force by one quarter well changing the

08:11

distance makes a big difference here when it comes to gravity you know just [Man in kitchen with butter gun]

08:14

like our butter gun man this thing is a scientific marvel and a bargain too - But

08:19

let's take a look at gravity in our own neighborhood well the earth of the Moon [Moon orbiting Earth]

08:23

are about 4 times 10 to the 8 meters apart from each other

08:27

it depends on the time of the year but that's about the average the earth has

08:30

mass of six times 10 to the 24th kilogram and the moon has a mass of 7

08:35

times 10 to the 22nd kilogram well what's the force of the earth pulling on

08:41

the moon? and what is the force of the moon

08:44

pulling on the earth all right wow so the moon is big really really big and [Moon spinning in space]

08:49

the earth is two orders of magnitude bigger than that we knew they wouldn't

08:53

exactly fit in the trunk of our car but still when you see the numbers you know

08:57

yikes alright let's make sure we have all the [Man discussing gravitation of the moon and earth]

08:59

numbers we need for our gravitational constant equation: big G is always there

09:04

because well it's a constant then we have the mass of

09:09

the earth and the mass of the moon those are our mass one and mass two and sure [Mass of earth and moon appears]

09:14

enough we've got our distances there so all we have to do is plug in the numbers

09:18

and since our masses and distance only had one sig fig our answer will too, which

09:24

gives us 2 times 10 to the 20th Newtons but wait there's more you knew we'd say

09:30

that is that the force that the earth is exerting on the moon or is that the

09:33

force the moon is exerting on the earth well the answer to that question is yes

09:38

it's both the earth and the moon are a third law pair here so they're doing that

09:43

whole equal and opposite thing on each other [Earth exerting gravitational pull on the moon]

09:46

it's like Newton is double dipping on this problem we've got his third law of

09:51

motion and his universal law of gravitation and we get it Isaac you're

09:56

awesome but did you would invent a better way to butter toast [Man speaking to Newton painting]

10:00

well did you smart guy? Alright, let's

10:05

look at one more problem and we'll keep it local all right like we just said the

10:09

moon and the earth are 4 times 10 to the 8th meters apart and the earth still has its

10:13

mass of 6 times 10 to the 24th given all this information how fast is the moon

10:18

orbiting the Earth? well hold on we're not even given the [Man discussing question on shmoop show stage]

10:22

moon's mass here come on how are we supposed to figure out its speed okay

10:26

well let's just mark down what we know right now and then we'll figure out how

10:29

to get to where we're going we're looking for a velocity and like always

10:33

we've got big G in the mix we've got the Earth's mass and we have a distance to

10:38

work with but our law of universal gravitation equation doesn't even have

10:42

velocity in it is there an equation that does use velocity well we've got

10:47

circular motion going on here and that's what an orbit is after all we know that [Moon orbiting the Earth]

10:52

circular motion involves a centripetal force and the

10:55

centripetal force equation uses velocity ding-ding-ding that equation says that

11:00

the centripetal force equals mass times the square velocity divided by the [Man discussing centripetal force equation]

11:04

radius well in the case of an orbit the centripetal force comes from gravity and

11:08

we have our universal law of gravitation equation for that force so can set these

11:13

two equations equal to each other like soand since the moon orbits the earth

11:18

we would use the moon's mass in our centripetal force equation after all [Two centripetal force equations appear]

11:23

it's the moon's velocity that we're trying to figure out all right now that

11:26

we've set these equations equal to each other we can cancel out the variables

11:30

that they have in common so we can get rid of the moon's mass altogether and we

11:35

can also kick out one R from each side too.. all right well now we're left with this [r unit removed from equation]

11:40

equation big G times the mass of the earth over the distance equals velocity

11:44

squared well do we have the right numbers for all those variables? yeah we

11:49

sure do we can plug in the numbers and solve for V when we multiply big G times [velocity equation appears]

11:54

the mass of the earth and divide that by distance and don't forget that we need

11:58

to find the square root well we come up with a velocity of roughly 1,000 meters

12:03

a second which is 3,600 kilometres an hour or about 2,237 miles per hour and

12:10

man everything in space moves crazy-fast I guess there's no point in stopping to

12:15

smell the roses when you're in a total vacuum so gravity in space are pretty [Astronaut floating in space]

12:20

big concepts to wrap our heads around I mean everything in the universe seems so

12:24

big billions and trillions and gazillions of kilometres between planets

12:28

and stars but at the same time all these planets and stars and asteroids and [Lines join between planets and stars in space]

12:33

everything are all pulling on each other thanks to gravity it's more than our

12:38

minds can really comprehend it especially when you haven't had a

12:41

balanced breakfast and what breakfast is balanced without a scrumptious piece of

12:46

buttered toast... just saying [Man fires butter gun at toast]

Full Transcript