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### Course: Physics library > Unit 4

Lesson 1: Circular motion and centripetal acceleration- Race cars with constant speed around curve
- Centripetal force and acceleration intuition
- Visual understanding of centripetal acceleration formula
- What is centripetal acceleration?
- Optimal turns at Indianapolis Motor Speedway with JR Hildebrand
- Calculus proof of centripetal acceleration formula
- Loop de loop question
- Loop de loop answer part 1
- Loop de loop answer part 2

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# Centripetal force and acceleration intuition

The direction of the force in cases of circular motion at constant speeds. Created by Sal Khan.

## Want to join the conversation?

- I'm still not able to digest the role of friction as a centripetal force in case of a car taking a turn. Doesn't it oppose the motion, rather than changing the direction? How does it actually function as a centripetal force? Looking forward to a helpful response.(21 votes)
- if a car in a racetrack is turning to the left(towards the center) its wheels will be turning to the right(away from the center)

https://qph.fs.quoracdn.net/main-qimg-cf054973c22da383a8d879e0bf11c3bb

the static friction allows the road to push back against the wheels of the car making it turn.

if there was no friction the car would slide straight or away from the center, to make it move in a circle friction must act inwards, so that's how it act as a centripetal force.(4 votes)

- is centripetal force a pseudo force?(9 votes)
- no not pseudo. Good question.

It is the name that we give to a force that causes an object to move in a circualr path. examples:

For the Earth around the Sun: Gravity IS the centripetal force

For a ball swinging around your head on string: Tension in the string IS the centripetal force.

Hope that clarifies(16 votes)

- what happens when you have uniform non zero acceleration(6 votes)
- I'm going to assume that you are questioning about Centripetal Force. The "Non zero" acceleration you are referring to is an acceleration that is either - acceleration or +acceleration but not 0 acceleration. If your meaning is - acceleration, the answer would be the car stays still since you cannot decelerate from initial velocity of 0 because the car is not applying force against gravity which is an conservative force and friction is an non-conservative force or if the car had some initial velocity, it would reduce its velocity to 0. If the car had some initial velocity and was decelerating, the centripetal force the tire supports would decrease and the centripetal acceleration would decrease since the car is slowing down. The centripetal force and acceleration changes since they only stay constant if the car's velocity was constant. Now, if you were referring to + acceleration, the friction of the tire would have to support more centripetal force and would increase since the velocity of the car is going faster. It would continue until the tire of the car cannot support enough force to turn it. The friction of the tire is given by the coefficient of kinetic friction times mass of the car times gravity or Greek letter mu (u with a line at the left)mg.

Hope this helped and your curiosity has been rewarded with a + 1 vote :)(11 votes)

- I have a doubt regarding the procedure used to find the direction of the ∆V vectors which is supposed to be in the direction of the centripetal force that is accelerating the body & changing it's direction at every instant. At2:10s Sal copies & pastes 2 velocity vectors having the same magnitude but having different directions. Let us assume the second vector is after time interval Δt from the first vector. So during that time interval it experiences an acceleration due to the centripetal force and hence changes it's direction by an angle Δθ. So now we have a new vector with the same magnitude but at an angle Δθ w.r.t to the first vector. Now when we try to find the ∆V vector by joining the tip of these two vectors which are co-centric but have a certain angle between them, then geometrically it is not possible for the supposed ∆V vector to be at right angle4:58to the initial velocity vector, or for that matter it can't be perpendicular to any of the two velocity vectors as is shown by translating the vectors at3:50. And if it is not perpendicular it is no longer pointing to the center of the circe!! To make the ∆V vector perpendicular to the initial velocity vector, the second velocity vector has to be larger than the first one in magnitude as per the pythagoras theorem. Then it would no longer be a case of uniform acceleration. It would then be a variable acceleration and the magnitude of the second velocity would be larger. There could be some problem with the logic given in this video or may be I am confusing something. Would appreciate if you could clear my doubts on this matter...Thanks(8 votes)
- You are right. If you compared the first velocity vector with a different velocity vector other than the second one, you will find that the change in velocity is not perpendicular to the velocity vector, and therefore the acceleration won't be perpendicular. But you must keep in mind that that is the direction of the average acceleration! If you take into account two velocity vectors at a very small time interval "dt" and took the change in velocity(dv), you will be able to convince yourself that the direction of "instantaneous acceleration"(which is what we want) is towards the centre. Therefore a=dv/dt acts towards the centre. Note that the "d" in the dv and dt means a very small change.(1 vote)

- So what IS the difference between centripetal force and centrifugal force?

If I'm understanding Sal's explanation correctly, centripetal force is a force that pulls object to the centre of the circle/gravity, then what is a centrifugal force?(2 votes)- Centrifugal force is an apparent force that comes from looking at something from a rotating frame of reference. For example when you are on a spinning marry-go-round you feel like you are being pushed away from the center. This force you feel if you describe it from the viewpoint of the marry-go-round it is an outward force the centrifugal force.

If you look at this same marry-go-round from the viewpoint of the ground the force on the person is seen as a inward force making them travel in a circle instead of a strait line this is the centripetal force.(13 votes)

- Why is it that @4:08he draws the white arrow from the arrow head but in the circle he draws the white arrow from the bases of the coloured arrows?(6 votes)
- It does not really matter. Whether he draws it from the arrow head or the base, the laws of vector addition still make them both equivalent. The point is that all those change in velocity vectors point toward the center of the circle, making it a purely rotational motion. They would all point to the center whether or not the "change in velocity vector" (Delta V) is drawn from the base of the velocity vector or from the arrow on the velocity vector.

Also, the diagram to the right in the video begins to set you up for free body diagrams for objects rotating around a point, in which all forces will be drawn from the center of mass of the object.(3 votes)

- So, would a satellite be considered a perpetual motion machine?(5 votes)
- The satellite only revolves around the gravitational body if the gravitational body exists. That's not perpetual motion. However, if the gravitational body were removed and the satellite revolved around nothingness, it's perpetual motion.(3 votes)

- What happens when the velocity isn't constant?(2 votes)
- Then that is when tangential acceleration comes into play. We have the centripetal acceleration changing the direction of the velocity vector and when the tangential acceleration changing the magnitude of the velocity vector.(3 votes)

- Why are the change in velocity vectors placed perpendicular to the velocity vectors?(3 votes)
- not always...

Only , in this case because the accelration (change in velocity / time) is towards the centre ofthe circle which is, at the same time, perpendicular to the (tangential) velocity(3 votes)

- What is the difference between Torque and Centripetel Force?(1 vote)
- Torque is a twisting force.

Centripetal force is a force toward the center of a circular path.(5 votes)

## Video transcript

Let's say we observe some object-- let's say for the sake of argument, it's happening in space It's traveling in a circular path with the magnitude of its velocity being constant Let me draw its velocity vector The length of this arrow is the magnitude of the velocity I want to be clear. In order for it to be traveling in the circular path the direction of its velocity needs to be changing So this time the velocity vector might look like that After a few seconds the velocity vector might look like this After another few seconds the velocity vector might look like this I'm just sampling. I actually could've sampled after a less time and it would be right over there but I am just sampling sometimes as it travels around the circle After a few more seconds the velocity vector might look something like that I want to think about what needs to happen what kind of force would have to act in particular the direction of the force would have to act on this object in order for the velocity vector to change like that? This remind ourselves if there was no force acting on this body this comes straight from Newton's 1st Law of motion then the velocity would not change neither the magnitude nor the direction of the velocity will change If there were no force acting on the subject it would just continue going on in the direction it was going it wouldn't curve; it wouldn't turn; the direction of its velocity wasn't changing Let's think about what the direction of that force would have to be and to do that, I'm gonna copy and paste these velocity vectors and keep track of what the direction of the change in velocity has to be Copy and paste that So that is our first velocity vector Copy all of these. This is our second one right over here Copy and paste it I'm just looking at it from the object's point of view how does the velocity vector change from each of these points in time to the next? Let me get all of these in there This green one That. Copy and paste it That. I could keep going, keep drawing velocity vectors around the circle but let me do this orange one right over here Copy and paste So between this magenta time and this purple time what was the change in velocity? Well, we could look at that purely from these vectors right here The change in velocity between those two times was that right over there That is our change in velocity So I take this vector and say in what direction was the velocity changing when this vector was going on this part of the arc It's roughly--if I just translate that vector right over here it's roughly going in that direction So that is the direction of our change in velocity This triangle is delta; delta is for change Now think about the next time period between this blue or purple period and this green period Our change in velocity would look like that So while it's traveling along this part of the arc roughly it's the change in velocity if we draw the vector starting at the object It would look something like this I'm just translating this vector right over here I'll do it one more time From this green point in time to this orange point in time and obviously we're just sampling points continuously moving and the change in velocity actually continues changing but hopefully you're going to see the pattern here So between those two points in time, this is our change in velocity And let me translate that vector right over there It would look something like that change in velocity So what do you see, if I were to keep drawing more of these change in velocity vectors you would see at this point, the change in velocity would have to be going generally in that direction At this point, the change in velocity would have to be going generally in that direction So what do you see? What's the pattern for any point along this circular curve? Well, the change in velocity first of all, is perpendicular to the direction of the velocity itself And we haven't proved it, but it at least looks like it Looks like this is perpendicular And even more interesting, it looks like it's seeking the center The change in velocity is constantly going in the direction of the center of our circle And we know from Newton's first law that if--the magnitude could stay the same but the velocity change in any way, either the magnitude or the direction or both there must be a net force acting on the object And the net force is acting in the direction of the acceleration which is causing the change in velocity So the force must be acting in the same direction as this change in velocity So in order make this object go in this circular there must be some force kind of pulling the object towards the center and a force that is perpendicular to its directional motion And this force is called the centripetal force Centripetal Not to be confused with centrifugal force, very different Centripetal force, centri- you might recognize as center and then -petal is seeking the center. It is center seeking So this centripetal force, something is pulling on this object towards the center that causes it to go into this circular motion Inward pulling causes inward acceleration So that's centripetal force causing centripetal acceleration which causes the object to go towards the center The whole point why I did this is that at least it wasn't intuitive to me that if you have this object going in a circle that the change in velocity, the acceleration, the force acting on this object would actually have to be towards the center The whole reason why I drew these vectors and then translate them over here and drew these change in velocity vectors is to show you that the change in velocity is actually towards the center of this circle Now with that out of the way, you might say, well, where is this happening in in everyday life or in reality in some way it perform And the most typical example of this and this is something that I think most of us have done when we were kid if you had a yoyo My best attempt to draw a yoyo If you have a yoyo and if you whip it around on a string you know that the yoyo goes in a circle Even though its speed might be constant, or the magnitude of its velocity might be constant we know that the direction of its velocity is constantly changing It's going in a circle and what's causing it to go in a circle is your hand right over here pulling on this string and providing tension into the string So there's a force, the centripetal force in this yoyo example is the tension in the string that's constantly pulling on the yoyo towards the center and that's why that yoyo goes in a circle Another example that you are probably somewhat familiar with or at least have heard about is if you have something in orbit around the planet So let's say this is Earth right here and you have some type of a satellite that is in orbit around Earth That satellite has some velocity at any given moment in time What's keeping it from not flying out into space and keeping it going in a circle is the force of gravity So in the example of a satellite or anything in the orbit even the moon in orbit around the Earth the thing that's keeping an orbit as opposed to flying out into space is a centripetal force of Earth's gravity Now another example, this is probably the most everyday example because we do it all the time If you imagine a car traveling around the racetrack Let's draw a racetrack. If I have a racetrack Before I tell you the answer, I'll have you think about it It's circular. Let's view the racetrack from above If I have a car on a racetrack. I want you to pause it before I tell it to you because I think it's an interesting thing think about It seems like a very obvious thing that's happening We've all experienced; we've all taken turns in cars So we're looking at the top of a car. Tires When you see a car going at a constant speed so on the speedometer, it might say, 60 mph, 40 mph, whatever the constant speed but it's traveling in a circle so what is keeping--what is the centripetal force in that example? There's no obvious string being pulled on the car towards the center There is no some magical gravity pulling it towards the center of the circle There's obviously gravity pulling you down towards the ground but nothing pulling it to the side like this So what's causing this car to go in the circle as opposed to going straight? And I encourage you to pause it right now before I tell you the answer Assuming you now unpaused it and I will now tell you the answer The thing that's keeping it going in the circle is actually the force of friction It's actually the force between the resist movement to the side between the tires and the road And a good example of that is if you would remove the friction if you would make the car driving on oil or on ice or if you would shave the treads of the tire or something then the car would not be able to do this So it's actually the force of friction in this example I encourage you to think about that