# Physics: Balanced Force Equations

Balanced force equations: the zen of Newton.

 Courses Physics Language English Language

### Transcript

00:39

balance in your life balance mind so you don't get too obsessed over stuff like [woman napping on her books]

00:43

physics a sense of balance so we're not falling down all the time and a balanced [model falls on cat walk]

00:48

breakfast it turns out that eating three bowls of Lucky Charms doesn't actually [woman pouring cereal]

00:52

set you up for a successful day and we know that balance is a big part of

00:57

physics too when we're dealing with constant velocity whether the velocity [woman on tight rope]

01:01

is positive negative or non-existent we know that we have balanced forces

01:06

whenever we have balanced forces we can set up balanced force equations in a [black board]

01:12

balance force equation we break down all the forces acting on an object to their

01:18

X&Y components then we can set up formulas showing that the force is

01:23

pointing to the left are equal to the forces pointing to the right and that

01:28

the force is pointing up are equal to the forces pointing down again this only

01:33

works when we don't have any acceleration happening if the velocity

01:37

is changing then we know we're dealing with unbalanced forces and all of this

01:43

balanced equation stuff goes out the window for now we're not going to put [man flies paper airplane]

01:47

any numbers into these equations it's more important to understand how these

01:52

equations get set up besides we're pretty sure you can do the actual math

01:57

when the time comes and we will run through one problem with actual numbers

02:01

at the end of the lesson once we do have magnitudes to work with we should still [kid sitting at desk]

02:06

set up the equations first before we start dealing with the values it'll help

02:11

keep things nice neat we like nice and neat especially

02:14

when it comes to math let's start off with something pretty simple your

02:19

average everyday Bulldog pushing a kid in a stroller we're sure you have [dog pushing stroller in park]

02:23

neighbors like these near you the stroller is moving at a constant

02:27

velocity from right to left so what does this tell us about the forces that are [stroller rolls past metal bars]

02:32

acting on the stroller let's look at this Freebody diagram first we'll draw

02:38

the line for the ground the square here will represent the stroller now let's

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think about what's going on for one thing we know we've got a slobbering [bulldog in park]

02:47

pooch providing applied acceleration in this direction we call that vector F sub

02:53

a and any time we have motion we know we've got friction acting in the

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opposite direction let's mark that F sub F and since the stroller isn't picking

03:03

up speed careening faster and faster down the sidewalk until it all ends in a [stroller crashes against building]

03:08

fury crash we know those forces must be balanced so to write it as an equation

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we can say that F sub a equals F sub F as always we've got gravity acting on

03:20

the stroller to which is the F sub G vector here and the ground is providing

03:26

an upwards normal force F sub n so we can say that F sub G equals F sub n but

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we can break down a little further we know that F sub G represents the weight [man punches board into two]

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of the stroller weight equals mass times gravity and since F sub n equals F sub G [numbers on black board]

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the normal force also equals mass times gravity so if we're given the mass of

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the stroller we could figure out the normal force by plugging in the

03:57

acceleration of gravity easy peasy when we've got balanced forces acting in

04:02

perpendicular directions it's not too hard to figure out our equations but

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things get a little bit trickier when we start adding angles into the mix let's

04:12

talk about the time the circus clowns had a little party after a show [clowns partying]

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things got pretty out of hand and when we came into the big top the next

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morning there were passed out clowns like everywhere we had to drag these [clowns passed out on floor]

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goofballs by their hair to get them to their trailers we'll say we're providing [woman drags clown away]

04:29

a constant velocity from left to right let's draw an FBD of this situation to

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figure out what forces are active will draw the clown as a square so we don't

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have to look at his painted face and have nightmares [black and white picture of clown face]

04:41

we'll start with gravity F sub G and the normal force F sub n pretty

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straightforward so far our motion is going this way so we know we've got

04:51

friction going the other way we'll add that in and call it F sub F what about

04:56

the force in the direction of motion though like we said we're pulling this [woman pulling clown across floor]

05:00

bozo by the hair meaning we're holding him at an angle

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and the force is really a tension force yeah it turns out the big red pouf isn't

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a wig it's the real deal so that force is at an angle we'll call that F sub T

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pen draw it like this hmmm S sub T can't balance friction

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because it's not in the same direction but using the X and y directions as the

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other two sides of a right triangle means that we can find the components of

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the triangle we just made we'll tackle the X Direction ie the horizontal first

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now theoretically we could just say that F sub F equals F sub TX but that

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wouldn't be really helpful would it if this was a real experimental situation [woman continues to drag clown across floor with doctor watching]

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we would be able to measure the force being applied by F sub T so let's figure

05:53

out how to really solve for F sub TX and we can do that with something called [numbers on black board]

05:58

trigonometry our angle of incline is this one right here will Papa theta on

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that going back to our trig functions which is the right one to find this side

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well if we use our old mnemonic sohcahtoa we know that the sine of an

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angle equals the opposite side over the hypotenuse the cosine of an angle equals

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the adjacent side over the hypotenuse and the tangent equals the opposite over

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the adjacent like we said in this experiment we'd be

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able to know the tension force which is the hypotenuse of the triangle and the

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horizontal side would be the adjacent of our theta angle so the right function is

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cosine and if the cosine of theta equals f sub t x over F sub t then we can

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rearrange the equation to solve for f sub t X to do that we multiply both

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sides by the hypotenuse F sub T so f sub TX equals F sub x the cosine of theta

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and F sub F equals the same thing - it's all about maintaining balance am i right

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as for f sub T Y it would be the same process for that but we'd used the sine

07:18

function instead of cosine we'd multiply both sides by F sub T to find that X sub

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T y equals F sub T times sine theta taking another look at our diagram we

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see two upward arrows F sub n and F sub T Y and the downward arrow F sub G since

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there's no acceleration vertically we know that these forces have to be

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balanced which means that F sub G equals F sub n plus F sub T why what does F sub

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G equal it equals mass times the acceleration of gravity so we can put

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that in place of F sub G now let's get really crazy what if we have a lion that

08:05

escaped from his cage and is sitting on one of the acrobats seesaws you see [lion on see-saw with lion trainer holding chair]

08:09

Pedro easy there here's what the Freebody diagram would look like we've

08:15

got our ramp here and gravity is always going straight down the normal force is

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perpendicular to the surface of the incline and friction is acting in the

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direction up the ramp we know that because otherwise is cutley ferocious

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beast but be sliding down the incline since he's staying still friction must [lion filing claws on rock]

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be holding him in place so here we have one completely vertical line and two

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agonal lines we could find the components for F sub N and F sub F but

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that would involve a lot of trig we like doing as little trig as possible so what

08:49

have we take a moment to warm up maybe crack our neck a couple of times and

08:51

look at this inclined from a tilt here we'll save you from any injury risk and

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do the tilting for you okay that change of perspective makes things a little

09:00

easier now we've got a diagonal line instead of two that cuts the amount of

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trig we need to do by half now we can draw the vector F sub GX parallel to the

09:11

incline and the wide component perpendicular to the incline now we can

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set up our balanced forces equation like so in the x-direction we have F sub F

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equal to F sub G X and in the Y direction that would be F sub N equals F

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sub G Y but when we're dealing with actual numbers that's not gonna give us

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everything we need if we're trying to figure out exactly how much the normal

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force is we're not going to be able to just say oh it's equal to F sub G Y that [people in a meeting]

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would be like saying like oh it's equal to purple it doesn't actually mean

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anything yet because F sub G y is just some line we drew we need to put it in

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terms of F sub G if we know how F sub G Y relates to F sub G then we can just

10:01

weigh Pedro the lion measure the angle and we'll have our answer because if we [lion on see-saw]

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know the length of one side of our triangle and the degrees of the non

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90-degree angles we can use trig functions to figure out everything else

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oh wait what angle are we talking about when we say to measure the angle we have

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a known angle here that's the angle of the incline does that match with [woman gesticulating]

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anything in this new triangle we drew well this hypotenuse is parallel with

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the opposite side in the Triangle created by the seesaw and this part F

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sub G is parallel with the hypotenuse of our

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starting triangle remember the trig functions like sine and cosine are all

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ratios so if an angle in a right triangle has a cosine of 0.5 that means

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the hypotenuse is twice as long as the adjacent side it doesn't matter if the

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hypotenuse is 2 meters and the adjacent side is 1 meter or if it's 80 meters for

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the hypotenuse and the adjacent is 40 the ratio is the same so what does that

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have to do with the whole lion situation we have here well in our mini triangle

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that we made out of f sub G and it's XY components we have two sides that are

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parallel to our big boy triangle since they're parallel that means their ratio

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must be the same as the corresponding sides of our big boar triangle so this

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angle here we'll call it a must be equal to the angle where the parallel sides of

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the triangle meet like bad this one right here will give it a really big a

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to match and since the three angles of a triangle add up to 180 degrees and we

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know two of the angles and each triangle are equal to each other that means the

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last angle in the triangles have to be equal to since we know our angle of

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incline at least theoretically we know that this angle at the top of

11:58

the mini-me triangle also equals theta whoo all this trick makes juggling while

12:04

riding a unicycle on a tightrope look easy so now we can use our sine and [woman juggling while riding a unicycle on a tightrope]

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cosine functions to put the component vectors for gravity in terms of F sub G

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sine of theta equals F sub G x over F sub G when we rearrange to solve for F

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sub G X we find that it equals F sub G times sine theta as for gravity on the

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y-axis we'll use cosine cosine of theta equals F sub G Y over F sub G which is

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once again rearrange to solve for F sub G Y F sub G y equals F sub G times the

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cosine of theta now put all of this into our balance force equation

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F sub n equals F sub G y we know that F sub n also equals F sub G times cosine

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beta last step Sub out F sub G for mass times gravity

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soo F sub n equals mass times gravity times cosine theta why is this important

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it's important because if we can weigh our Lian and find the angle of the [lion on a scale then man measures see-saw]

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incline we can figure out the force of friction and the normal force yep just a

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scale and a protractor and we can find the forces of the universe well a couple [picture of galaxy]

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of them at least pretty cool right let's look at one more problem and actually [woman juggling]

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use some numbers this time one of our clown's biffo has shaken off that rough [clown rides bike up platform]

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night and is biking up a ramp to a diving platform he's going to do one

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heck of a belly flop but let's figure out some stuff before he gets that far

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we'll say he's exerting 900 Newton's of force to climb up this ramp which is at

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a 35 degree angle bippo and his bike have a mass of 75 kilograms assuming

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he's moving at a constant velocity what is the force of friction and the normal

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force of the ramp okay here's our FB D whoa yikes three diagonal angles to deal

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with okay let's do that whole tilty thing to make this problem easier okay

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much better now we just have one diagonal to deal with we'll sketch in

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the X and y components of gravity and like we just saw before our known angle

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theta will go at the top here as the angle between F sub G and F sub G Y

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Biffle is moving at the same speed so there's no positive or negative

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acceleration so the two vectors pointing backwards from bibo add up to equal the

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vector in the direction of his motion F sub F plus F sub GX equals F

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a we want to solve for F sub F so we can subtract F sub G X from both sides to

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isolate our target oh yeah now we're making progress there's no acceleration

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along the y axis either so the arrows along that plane are also equal F sub n

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equals F sub G Y sticking with this vertical stuff

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let's get F sub G Y into terms of X sub G like we said previously the cosine of [stock market on market street screens]

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theta equals F sub G Y over F sub G rearranging that to solve for F sub G Y [numbers on black board]

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shows that it equals F sub G times cosine of theta one last step before we

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start putting in the numbers let's swap F sub G for mass times gravity after all

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they're the same thing so the normal force equals mass times gravity times

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the cosine of theta all right number time in case you forgot Pipo and his

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bike have a mass of 75 kilograms and the theta is 35 degrees gravity is the same

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old 9.8 m/s^2 so the normal force equals 75 kilograms

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times 9.8 meters per second squared times cosine of 35 degrees when we

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multiply all those together we find a force of about 602 Newton's or 6.0 times

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10 to the 2nd to follow the significant figures rules

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look at those clown legs pump away little bit I'll go okay onto the [clown riding bike up ramp]

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x-direction to find our friction we've already figured out that the force of

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friction equals the applied force minus the X component of gravity let's get

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that X component into terms of F sub G will use sign for that sine theta equals

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F sub G x over F sub G solving for F sub G X we can rearrange the equation so

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that F sub G X equals F sub G times sine theta okay we'll put that into the

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balanced force equation we have for F sub

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F sub F equals F sub a minus F sub G times sine theta and just like we did in

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the y-direction we can step out F sub G for mass and gravity biffo is applying

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900 Newton's of force on his way to the diving platform so when we plug in the

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number our equation looks like this F sub F equals 900 Newton's minus the

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product of 75 kilos times 9.8 meters per second squared times the sine of 35

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degrees our calculator tells us that the force of friction equals four hundred

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seventy eight point four Newton's using the right number of sig figs which in

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this case is two gives us a result of four point eight times ten to the second

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Newtons after all this your belly-flop had [clown falls off ramp into pool]

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better be worth it clown okay okay not bad fellas thing seems pretty easy when

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you're just standing on flat ground but once you start adding other forces or

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angles or flaming torches balancing can get a lot more complicated just remember [woman juggles flaming torches while riding unicycle on tightrope]

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to take everything step by step and set up your equations before you start

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worrying about any actual numbers and even if you make a wrong step and lose [woman walking on tight rope then falls]

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your balance it's okay you have the luxury of being able to start over again

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as for us well because hope a clown will break our fall [woman falls on clown]