How fast can a racecar travel through a circular stomach without skidding and hitting the wall ? The answer could depend on several factors :
- The weight of the car;
- The friction between the tires and the road;
- The radius of the circle;
- The “steepness” of the turn.
In this stick out, we investigate this interrogate for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this .
A cable car of mass megabyte moves with changeless angular speed to around a circular curve of radius R ( Figure 3.20 ). The swerve is banked at an lean . If the stature of the car off the background is planck’s constant, then the stead of the car at time metric ton is given by the function
As the car moves around the curl, three forces act on it : gravity, the power exerted by the road ( this force out is perpendicular to the footing ), and the friction pull ( Figure 3.21 ). Because describing the frictional force generated by the tires and the road is complex, we use a standard approximation for the frictional force. Assume that for some positive constant . The changeless is called the coefficient of clash .
Let denote the maximal speed the car can attain through the wind without skidding. In other words, is the fastest focal ratio at which the car can navigate the turn. When the car is navigate at this amphetamine, the order of magnitude of the centripetal force is
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The next three questions deal with developing a formula that relates the speed to the banking angle .
immediately that we have a formula relating the maximum speed of the car and the banking angle, we are in a position to answer the questions like the one posed at the beginning of the project .
The Bristol Motor Speedway is a NASCAR short track in Bristol, Tennessee. The track has the estimate supreme headquarters allied powers europe shown in Figure 3.22. Each end of the track is approximately semicircular, so when cars make turns they are traveling along an approximately circular curve. If a car takes the inside track and speeds along the penetrate of turn 1, the car travels along a semicircle of radius approximately with a bank angle of . If the car decides to take the outside path and speeds along the acme of turn 1, then the cable car travels along a semicircle with a banking angle of . ( The track has variable angle bank. )
The coefficient of friction for a normal fire in Elly conditions is approximately . therefore, we assume the coefficient for a NASCAR tire in dry conditions is approximately .
Before answering the following questions, note that it is easier to do computations in terms of feet and seconds, and then convert the answers to miles per hour as a final step .
11. Suppose the measure amphetamine of a car going along the outside edge of the turn is . Estimate the coefficient of friction for the car ’ s tires .