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23 Nov 2019

You have a car traveling on a circular track of radius r and avelocity v. The
track is banked at an inclination of angle . The coefficient ofstatic friction
is µs. Gravity is 10m
s2 . In each of the following 5 situations, does the car:
• A. Fly off the top of the track.
• B. Slide down and off the bottom of the track.
• C. None of the above.

1. = 45?, r = 1000m, v = 90m
s , µs = 0.5
2. = 20?, r = 1000m, v = 90m
s , µs = 0.2
3. = 70?, r = 800m, v = 200m
s , µs = 0.7
4. = 30?, r = 800m, v = 5m
s , µs = 0.3
5. = 30?, r = 800m, v = 78m
s , µs = 0.3
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Related textbook solutions

Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

Related questions

1. An object of mass 19 kg is placed on incline with friction.The incline is originally horizontal and then raised slowly and at21 degrees, the mass begins to slide down the incline. It is foundthat the coefficient of kinetic friction is 0.79 times thecoefficient of static friction. If the same object (same mass) isplaced on the same incline with an inclination angle of 42.2degrees (the mass will slide down the incline at this angle), whatis the magnitude of the acceleration on the mass in m/s2?
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5. A roller coaster cart of mass 15.8 kg travels on the inside of aroller coaster track in a loop of radius 23.1 meters. The circle isa vertical circle similar to the yo-yo example done in class. Ifthe cart travels at a constant speed around the track and the valueof this speed is the minimum speed such that cart does not fall offthe track at the top of the track, was is the magnitude of thenormal force on the cart at the bottom of the track in Newtons?

How fast can a racecar travel through a circular tum 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 project, we investigate this question 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 car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function

 

As the car moves around the curve, three forces act on it: gravity, the force exerted by the road (this force is perpendicular to the ground), and the friction force (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 constant is called the coefficient of friction.

Let denote the maximum speed the car can attain through the curve without skidding. In other words, is the fastest speed at which the car can navigate the turn. When the car is navigate at this speed, the magnitude of the centripetal force is

 

The next three questions deal with developing a formula that relates the speed to the banking angle .

Now 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 approximate shape 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 bottom of turn 1, the car travels along a semicircle of radius approximately with a banking angle of . If the car decides to take the outside track and speeds along the top of turn 1, then the car travels along a semicircle with a banking angle of . (The track has variable angle banking.)

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 measured speed of a car going along the outside edge of the turn is . Estimate the coefficient of friction for the car’s tires.

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