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## Understand independence of component vectors, add vectors mathematically

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Banked With No Friction

### Banked With No Friction

Credit: susi.bsu
Source: http://www.flickr.com/photos/62517473@N06/5948763278/

At the Indianapolis Motor Speedway there are 4 turns that have the tracked banked at nine and twelve degrees. Compared to a flat track, inclined edges add an additional force that keeps the cars on their path. This force prevents the cars from being pulled towards the center or pushed out.

#### Why It Matters

• When a car is driving on a flat surface in a straight line, the following 4 forces are acting on it:
1. the weight of the car
2. the normal force
3. the force applied by the tires
4. the frictional force
• Credit: Hideki Kimura
Source: http://commons.wikimedia.org/wiki/File:Solar_car_race_suzuka.jpg

Cars driving in a straight direction experience four different forces [Figure2]

• These 4 forces keep the car moving in a straight line. There are no forces that are perpendicular to the other two forces that would allow the car to turn in a circular path.
• If a car is on a banked turn, centripetal force is an additional force on the car. Usually the normal force just has a vertical component, but in the case of a banked curve it has both a vertical and a horizontal component. In the case where friction exists, it must be added to the horizontal normal component since the frictional force points towards the center of the arc.
• Watch what happens when friction does not exist:

#### What Do You Think?

Using the information provided above, answer the following questions.

1. The equation for the maximum velocity of a car on a banked curve is given as: v=rg(tanθμs)1+μstanθ\begin{align*}v=\sqrt{\frac{rg(\tan \theta - \mu_{s})}{1+\mu_{s} \tan \theta}}\end{align*}. What does this equation simplify to when there is no banked corner?
2. Why does the normal force have a horizontal component when the car is on a banked curve?
3. What does the equation for the maximum velocity of a car on a banked curve simplify to when there is no friction?

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