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A) Look at the picture given on the right Drive an expression for Ue maximum specd of' the car (Vrz) in terms of thc radius 0[ the curve (R) the gravilation...

Question

A) Look at the picture given on the right Drive an expression for Ue maximum specd of' the car (Vrz) in terms of thc radius 0[ the curve (R) the gravilational acceleration (g} and the cocfficient frietion (0); so thal thc car does nOl fly off the road_ b) Walch lhe relaled parlS 0[ the YouTube vidco litled as "Formula Car Looks Unbelicvablc Under Thermal Imaging" from the YouTube channel *b/6O" According to the video_ and the formula you dcrivcd IJ parl #} make comment 0n

a) Look at the picture given on the right Drive an expression for Ue maximum specd of' the car (Vrz) in terms of thc radius 0[ the curve (R) the gravilational acceleration (g} and the cocfficient frietion (0); so thal thc car does nOl fly off the road_ b) Walch lhe relaled parlS 0[ the YouTube vidco litled as "Formula Car Looks Unbelicvablc Under Thermal Imaging" from the YouTube channel *b/6O" According to the video_ and the formula you dcrivcd IJ parl #} make comment 0n why Formula pilots pcrlorm lelt-right mancuvers during warm up laps.



Answers

The aerodynamic resistance to motion of a car is nearly proportional to the square of its velocity. Additional frictional resistance is constant, so that the acceleration of the car when coasting may be written $a=-C_{1}-C_{2} v^{2},$ where $C_{1}$ and $C_{2}$ are constants which depend on the mechanical configuration of the car. If the car has an initial velocity $v_{0}$ when the engine is disengaged, derive an expression for the distance $D$ required for the car to coast to a stop.

According to the given problem. I can write a question asked, happen is equal to m omega. Also I can like kmt e greater than equal to M B squared by our So the is smaller than equal to underwrote K R and D. So be max is equal to under route. He uh minimum G on solving it for the we know that radius of curvature for a car at any point tax why is given by R is equal to Model of one plus divide by dx squired All to the Power three x 2. Bye. The square by by dx square. So for the given carve, I can write the value of divide by the X is equal to a by alpha cause act by alpha and the square by by dx square is equal to minus a by alfa square signed X Y el palm on solving it further, I can also write the value of R is equal to one plus a square by Alfa Square bosses square act by Alpha. All to the power three x 2 by everybody. Alpha is square. Sign. Acts by Alpha. For minimum value of our odd minimum value of art. That's viol five is equal to five x 2. So corresponding radius of curvature can be written edge. Our minimum is equal to I'll play square by a mx is equal to who tended kg by a multiplication. Al part

Okay, so this problem is asking us to find the coefficient of friction that is necessary between the tires of this Formula one car and the road. So here you can see my not so great drawing of what's supposed to be a Formula one car. But regardless, we've got friction on each tire and we've got the normal forces and one and two on each tire as well. And we also have the weight of the car, The downward force applied 0.35 F on the front tire, 0.4 on the back tire and the remainder in the center of gravity, which also has the drag force affecting it. And so now we want to apply the conditions of equilibrium in the horizontal direction. So the first equation will get is a few times and one plus end too Plus 4000, which is the drag force equals 690 times four g. 690 is the mass of the car and the driver And 4G is the maximum deceleration of the car. And all three of these values are given. And so we can simplify this into an expression for N one and N two. If we subtract the 4000 over and then we divide by μ were able to get N one plus N two Equal to 23,000 in 75 0.6. And this is divided by mu Okay. And so next we want to do the same thing but this time in the vertical direction, so you've got N one and N two pointing upwards, that's equal to the force coming downwards which is F. And so the reason this problem might seem a little trickier than it really is is because we have here 0.35 F 0.25 F. And 0.4 F. All distributed in different parts of the car. But when we take the some of the vertical forces, these are all going to add up to just one F. Which is what we get here plus MG. Which again we know the values for these. And so now what we can do is substitute this value for N one plus end too substituted in here, and we can get 23 1075.6 divided by mu and that's equal to F plus MG, which is just the right hand side of this equation. And now we know F the downward force, we know MG. And we can solve from you the coefficient that we want, and this will come out to Approximately 1.167. So this is the coefficient of friction required to keep the car on the ground.

So from example 5.15 off our textbook, we see that angle theta is equal to 10 in verse. The not squared over r G Ah, talk about what those parameters mean. First, let's try the free body diagram off the car, which is on the bank. So that red looking box is the car. Um, so the angle or this is called the banking angle associated with the diagram is Dannon was we don't squared over r G where v notch, Where is tthe e speed? Uh, quiz. The car's moving Our is the radius. So basically, by doing this, banking will have some centripetal acceleration directed towards the center and, ah, let's say the direction will be towards the radius. And if we take the parallel direction to this as our X axis, so we'll say it's towards X axis and we take that as positive. And, jeez, the gravity which is acting $3 words. So let's start with the free world diagram associated with this situation. The first of all, we have md, which is acting downwards. Then we have the normal force, which is perpendicular. Tow the slope. We call it F off end. So by using trick, we see that if it makes an angle theater with the vertical and, ah, then there will be two components off FN one along. Why access and why excesses vertically upwards before its paws. We call it a positive ID. So what we learned from here is ah, if the car if we want to know the maximum speed of the car or the speed beyond which the car will start a skid on the upward direction uh, we'll see that friction will be applied down the slope because the car will have a tendency to go upward direction while skidding, so the friction will be in the opposite direction. We call that fo fart. So again, by using trickery, see that this friction forces making angle theta x. So we're all set with the people diagram. Now we need to do is solve for the Newton's second law, so we have to access when his effects and one is f y. So let's soul for F y first, so f y in every direction. We have the component of F n, which is, um, fo Franco Science data. That's a positive because we have already mentioned that upward Why direction is positive, but then mg and component off friction along mg is negative because they're in the opposite direction. So we have negative mg minus it sounded far saying and we said that equal to zero from here, we if we solve for f n we see that effin will be mg over course I am Data minus mu s signed later. How do we get, uh, this Mewes component here? So notice one thing. If when the car is on the verge of skidding, we said the frictional force equal doom us or coefficient of static friction times the normal force. So that's what that's what brings me you in our problems or what we did here is we substituted if somebody far back here and from there we see that we have a component FN or we have a term fn there. So if we take a friend outside And so for that putting effort left, we are end up with this equation now similarly for FX, we have if our which is the centripetal force and this is the force that's responsible for the sentimental emotion or the car that car is making. So this must be equal toe all components in the extraction. We see that effin has ah offend co sign Leda on that direction. And then we add that to the component of friction which is itself a far Costa Rica. Ah, One mistake here. This is not co signed. This ist ein because co sign is along. Why access? So this will be signed and this whole thing must be equal toe m times send relaxation a onda we know that is we squared over r where these the, um, angular speed or Vietnam Sorry V's d tendency ls feet are is the radius anarchist. And from here, if we against all for f n we use this controlling condition over here as well. You see that f off and is equal Do mg divided by co sign later minus new s signed data. So I wrote down the same expression. Um, So it's gonna be and v squared over r divided by scientist, huh? Plus from us, cause I'm better Sorry about the mistake. So we have two expressions for FN, and all we gotta do is equate them together And ah, Saul. For Mueller, it's all for the max. So this velocity is the maximum velocity backs among allowable velocity. So if we said them equal to each other, we have m v squared by are divided by sign Leda plus Mu s co signed data. It must be equal toe n g. Divided by course. I ain't Ada minus. I'm us signed data. And from here we see that we can get it off the EMS and finally fits all for V. It's no Call it be Max. So the max must be equal to the square. Root off our G sign. Trade up one Plus I'm us ready by tan stayed up divided by call Sign beta one minus us Tansy. We put this under square, dude and that can be simplified as the zero square root of one first rg us divided by p zero squared. Um, where visa is the normal speed that the car is traveling where the fiction is zero linus us be zero squared, divided by rg. So we we got this expression from here where the days equal to 10 in verse we not squared over r g where the note is the velocity on the friction that surface. All right, so that's the maximum velocity. What about the minimum velocity? So when the car is driving at a slower speed, what happens? Store diagram. So let me drive. It will draw it one more time. So we have stayed out here. Um, we have the car over here, and ah, then we have friction. I sorry. We have my normal force along the perpendicular direction to the slope. Then we have mg straight downwards. And ah, what about the access? We have X to the right and why in a poor direction. So in this case, while the car is going at a minimum speed, it will have a tendency to skid towards skit downwards or down the slope. So that means the friction must be acting on the opposite side, which is up the slope. We call it f off afar, and that will give us the minimum speed. So by trick, we see that this fr is making angle. Frieda, with the the horizontal direction. And ah, also, FN is making angle data with the vertical. So we'll do the same thing with will apply Newton's second law. In this case, we see that we have f why? And you know, for if I direction we have a Franco science data. So that's the component of a fennel way. Access minus mg because mgs in downward. But then we have a plus year because effort far is now in a poor direction. So it's f sub fr sign beta, and this whole thing must be with zero because there's no most motion in the vertical direction. And from here we see that seven is equal Do mg It's over cause I knw Leda plus new s scientist again, we're taking the condition as f sub fr is equal to Mu s ab seven and putting it back here similarly, in the extraction sub X is equal to it's a bar which is steps up and sign data minus Epps up. If our cool science data that's equal to M v squared over R, notice that the scientists change now instead of positive, this has been negative. So from here again, if you drive us all for the normal force, this becomes m b squared over ours, divided by sign data minus mu s coastline data. And again we should match these two expressions. We got, uh if offer if I replaced by US event and then if we mast east too we see that we have n v squared over r by scientists minus us Close ended up which is equal to m g over co sign plus me aside data So in full of the same calculation again. But now the differences instead of plus and minus we have minus and plus so that gives us the minimum velocity. And ah, we said them together. We see that the minimum velocity is ah, the one that we calculated just now. And the beam axes basically the same. The only change change is the science s o. We have ah, positive in the numerator and negativity nominated, which is Ah, of course, opposite for the minimum velocity. Thank you.

In this situation we have given a circular restrict Of Radius 300 m and it is banked at an angle of 50°. We have given the coefficient of friction That is .2 and we have to find the optimum speed of the race card to avoid wear and tear of its tire. So we know that institution, we have given data that is radius R. Is equal to 300 m and the angle of bank that is treated as equals to 15 degrees Gs given was that is 10 m per second squared. And the coefficient of static friction at this point to we can say that the optimum speed of God who award we are in theories that is physical to undercut of R. G. 10 to now put all the data in this situation. So we can say this is given by on the road of 300 into disease. Turn into 10 15. So this will comes out to be under root of 810 dot is nine road to. So it is a nine route 10 m/s. So this is the answer of argumentation and for that we can see option B is the correct choice. Thank you


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