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Teachers & SchoolsBalanced force equations: the zen of Newton.

Courses | Physics |

Language | English Language |

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

keep things nice neat we like nice and neat especially

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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]

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

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

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

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

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

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

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

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

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

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

sub G is parallel with the hypotenuse of our

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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]

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

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

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

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

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

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

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

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

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

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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

equals F sub G Y sticking with this vertical stuff

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]

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

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

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

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

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

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

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

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

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

10 to the 2nd to follow the significant figures rules

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

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

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

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

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

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

balanced force equation we have for F sub

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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