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A banked curve of radius $R$ in a new highway is designed sothat a car traveling at speed $v_{0}$ can negotiate the turn safelyon glare ice (zero friction). If a car travels too slowly, then itwill slip toward the center of the circle. If it travels too fast,it will slip away from the center of the circle. If thecoefficient of static friction increases, it becomes possible fora car to stay on the road while traveling at a speed withina range from $v_{\min }$ to $v_{\text { max }}$ . Derive formulas for $v_{\text { min }}$ and$v_{\text { max }}$ as functions of $\mu_{\mathrm{s}}, v_{0},$ and $R.$

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$$v_{\min }=v_{0} \sqrt{\frac{\left(1-\mu_{s} R g / v_{0}^{2}\right)}{\left(1+\mu_{s} v_{0}^{2} / R g\right)}}$$$$v_{\max }=v_{0} \sqrt{\frac{\left(1+\operatorname{Rg} \mu_{s} / v_{0}^{2}\right)}{\left(1-\mu_{s} v_{0}^{2} / R g\right)}}$$

Physics 101 Mechanics

Chapter 5

Using Newton's Laws: Friction, Circular Motion, Drag Forces

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Cornell University

University of Michigan - Ann Arbor

University of Washington

University of Sheffield

Lectures

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In physics, a rigid body is an object that is not deformed by the stress of external forces. The term "rigid body" is used in the context of classical mechanics, where it refers to a body that has no degrees of freedom and is completely described by its position and the forces applied to it. A rigid body is a special case of a solid body, and is one type of spatial body. The term "rigid body" is also used in the context of continuum mechanics, where it refers to a solid body that is deformed by external forces, but does not change in volume. In continuum mechanics, a rigid body is a continuous body that has no internal degrees of freedom. The term "rigid body" is also used in the context of quantum mechanics, where it refers to a body that cannot be squeezed into a smaller volume without changing its shape.

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In physics, rotational dynamics is the study of the kinematics and kinetics of rotational motion, the motion of rigid bodies, and the about axes of the body. It can be divided into the study of torque and the study of angular velocity.

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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.

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