ShmoopTube

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circular motion round and round around we go where we stop

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well physics probably knows....

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Well when you do crazy things [Man flying a plane]

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at an airplane it's good to have a decent understanding of physics like

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when you're flying upside down it's helpful to understand what you need to

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do to counteract the acceleration of gravity otherwise landing the plane [Plane landing upside down]

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could come a bit you know earlier than you were hoping for a bit rougher - same

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thing if you're gonna pull off a perfect loop-de-loop well you've got to know

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what's creating your centripetal force so you know how to make it work in your

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favor because no one likes a loop-de-loop that turns out to be a [Plane crashes into the floor]

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know loop-de-crash especially the pilot so how does circular motion really work

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well the key thing is centripetal force that's the force that's able to overcome

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an object's inertia and create the circular motion well it turns out [Plane travels in a circular motion]

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there's a handy little equation for centripetal force and it's this one

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right here this equation says that centripetal force F sub C equals mass

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times the square of the object's velocity divided by the radius of the

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circle got it okay and this works for whatever the centripetal force is

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whether that's tension, gravity or you know whatever and which comes in handy

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when we know the equation for whatever type of centripetal force we're dealing [Equations for types of centripetal force]

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with if the centripetal force is a friction

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force for example we know that the equation for friction is the coefficient

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of friction times the normal force so we have two equations for the same force we

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set them equal to each other and use them to find values for unknown

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variables now one of the basic laws of physics is that force equals mass [Formula for force]

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times acceleration well Isaac Newton dropped his three laws of motion way [Isaac Newton underneath a tree]

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back in 1687 and this equation was the basis of law number two because of this

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law we can say that centripetal acceleration equals velocity squared

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over the radius why well because everything is contained in the force

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equation except the mass and there is always acceleration when it comes to [Moon orbiting the Earth]

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circular motion even when the velocity is constant

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remember velocity is the rate of displacement over time and it's a vector

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quantity so it has both a magnitude and a direction acceleration is the rate of

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change in velocity over time and in circular motion even if the magnitude of

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velocity stays the same, the direction is constantly changing which means there's

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constant acceleration even when the speed stays the same it's kind of like a

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trick question of nature....

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yeah circular motion can definitely make your head spin [Mans heading spinning in circular motion]

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Now all this time we've been talking about centripetal

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force like it's just one thing but there can be more than one force creating the

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total centripetal force say we're planning with one of these cool toy [Man holding toy airplane]

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airplanes well we tie some string around it and

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spin it around vertically and wee boy! this is fun who needs an Xbox but let's

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freeze it here at the top all right at this point what is the centripetal force [Airplane frozen at the top of the circular motion]

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well we've definitely got tension on the string otherwise the toy plane here

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would just keep going parallel to the ground so the tension on the string is

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pointing to the center of the circle like any centripetal force does but

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there's another force pointing that way too that would be our old pal gravity so

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gravity and tension are working together to create the centripetal force in fact [Gravity and tension merge together]

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we could even write this as an equation like this one - gravity plus tension

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equals mass times velocity squared over the radius it's the opposite situation

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when the plane is at the bottom of the circle here tension is pulling up toward [Arrow showing tension pulling upwards]

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the center of the circle but gravity is still pointing down which means it's

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pointing away from the center it's like a bad relationship you know

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one minute they're working together the next they're in complete opposition and [A man and woman cuddling]

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it just keeps going in circles and so at the bottom of the journey the equation

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for centripetal force looks like this tension - gravity equals that

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centripetal force equation which means that the tension in the string is having [Man holding a string]

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to do a lot more work at the bottom of the circle than it is at the to

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Now that we have some equations to work with we can start doing some actual

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math... well back when I was just a little co-pilot my favorite thing at [Young boy wearing pilot gear at playground]

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the playground was the merry-go-round you'd get that thing spinning so fast it

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felt almost like flying... so let's say I put my son on one of these

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contraptions he loves it the kiddo has a mass of 10 kilograms and [Son stood on a merry-go-round]

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I get this thing spinning at a velocity of 5 meters per second it's pretty fast

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if the radius of the merry-go-round is 5 meters

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What's the kid's centripetal force and the centripetal acceleration....

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well let's start with the centripetal acceleration since that's a component of

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centripetal force like we've got to get that number first well the equation for

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this acceleration is velocity squared over the radius and with a velocity of 5

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meters a second and a radius of 5 meters we come up with an acceleration of 5 [Formula for son's centripetal force]

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meters per second squared to find the amount of force we can just multiply the

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acceleration by the mass which was 10 kilograms making the force 50 Newtons

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hope the little tyke holds on tight there well flying yeah is in his blood [Boy falls off the merry-go-round]

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Well after I convinced Jr. to basically never tell his mom about this

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we'll head off to get some ice cream yeah...The landscape around here is a

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little weird lots of hills and valleys in there almost like half circles well [Car on top of a hill]

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let's draw a force diagram for the car at the top of the hill and the bottom of

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the valley all right then we can figure out what our centripetal force equations

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would look like okay here we go so let's start at the top so we're in a moving

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car which means we have the applied force from the engine pushing us forward

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and friction is pushing in the other direction and how about along the y axis

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well there's definitely gravity in the normal force to deal with but are they

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equal or is one bigger than the other this is circular motion here people so [Circular motion appears at the top of the hill]

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there has to be a centripetal force the center of the circle is straight down

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and guess which direction gravity works yeah, that's our centripetal force

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but we've also got the normal force of the road pushing up because we're

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continuing in our circular motion there has to be acceleration right and in

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order for there to be acceleration well we have to have a net force. If the

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forces were equal in the net force was zero our motion would continue in the [Car flys across the hills]

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direction of the velocity vector and we'd be catching air on this hill with

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the tyke in his car seat back there I figure I should take it easy [Man in the car with his son in the backseat]

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you know the straps aren't that tight...So gravity has to be stronger than the

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normal force here we'll show that on our diagram by making the gravity arrow

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longer than the one for normal force - now when we're at the bottom of the hill in [Car travels to the bottom of the hill]

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the valley here the situation's reversed we've still got our applied force and

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our friction but now the center of the circle is straight up just like the

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normal force and the normal force has to be greater than gravity so it'll get the

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longer arrow this time well you might have experienced this exact same thing [People riding a rollercoaster]

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on a roller coaster think of going down that first big hill well at the bottom

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you're pressed hard down into the seat that's because your inertia and gravity [Arrows point to inertia and gravity of people on a rollercoaster]

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are working in similar directions your body wants to keep going in the same

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downward angle of the hill and gravity is pushing you straight toward the

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ground so as you're being pushed down harder into your seat your seats pushing

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back up onto you with equal force so now all we have to do is figure out the [Car travels up the hill]

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equations for centripetal force at each point in our ice cream trip... At

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the top gravity minus normal force equals centripetal force and at the

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bottom it's normal force minus gravity and at the ice cream shop it's vanilla [Man in the car at the ice cream store]

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plus brownie fudge equals young yeah now sometimes Jr. gets bored so I

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brought along that plane on a string that I was playing with earlier well [Man gives Jr a toy plane]

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hopefully that'll keep him distracted long enough for me to finish the rest of

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his frosty treat as he's swinging the plane around vertically let's do [Jr swining plane in garden]

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something different at the top of the circle the plane has a velocity of 3

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meters a second it has a mass of 0.5 kilograms and the force of tension on

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the string is 15 Newtons let's put those numbers into a centripetal force

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equation and use that to find the radius of the circle hmm well this is gonna be

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a little trickier than what we've had to do before but it's nothing we can't

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handle we have the toys mass and its velocity and we know the tension force

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but we can't just plug all that into the centripetal force equation and solve for

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R because tension isn't the only force that's in play here. At the top [Jr holding toy plane on a string and gravity line appears]

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of the circle we've got gravity to factor in too - so our centripetal force

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equation will be gravity plus tension equals mass times velocity squared over

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the radius well since F sub G equals mass times the

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acceleration of gravity we can sub in those variables the equation gravity

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probably hasn't changed in the last few minutes so it's still 9.8 meters per

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second squared multiplying that by point five kilograms gives us a force of

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gravity of 4.9 Newtons and now we can plug in the numbers and find the radius

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well 15 Newtons plus 4.9 Newtons equals 0.5 kilograms times the square of 3 [Formula to find the radius on a board]

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meters per second over R when we do that first bit of addition, well we get

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19.9 Newtons and rearranging that equation to solve for R [Equation rearranged to solve for R]

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we find that R equals 0.5 kilograms times the square of 3 meters per second

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over 19.9 Newtons and Mr. calculator tells us that the radius then

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equals 0.23 meters it's a pretty tight turn there but we've got to use the

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right amount of significant figures so the radius is 0.2 meters not

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very big but big enough to do damage and that's what happens when you lose your [Boy swining toy plane on a string]

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centripetal force okay time to go it's important to recognize when there are

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two forces in action that are combining to create centripetal force or when

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they're in opposition with one creating the centripetal force and one acting

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against it and real life isn't always just up or down there might be times

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when we have to break a diagonal force vector into x and y-components but we [Blue ball travelling in circular motion]

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have all the skills now to handle that and even if we crash and burn on a

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physics problem well there are worse ways

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to crash and burn.. [Man parachuting down to the ground]

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