An engineer wants to design an oval racetrack such that 3.20 x 10³ lb racecars can round the exactly 1000 ft radius turns at 103 mi/h without the aid of friction. She estimates that the cars will round the turns at a maximum of 175 mi/h.

a) Find the banking angle [tex]\theta[/tex] necessary for the race cars to navigate the turns at 103 mi/h without the aid of friction.

b) What additional radial force is necessary to prevent a race car from drifting on the curve at 175 mi/h?

The banking angle θ for a race car to navigate a turn at 103 mi/h without the aid of friction can be calculated using the formula tan(θ) = (v2)/(gr). To find the additional radial force required at 175 mi/h, we use the centripetal force equation and subtract the force provided by banking.

Explanation:

The first step is to convert all the quantities in uniform units. So, convert the speed from mi/h to ft/sec for convenience. Using the formula for the banking angle for a car rounding a turn without the aid of friction:

tan(θ) = (v2)/(gr)

where 'v' is the speed (103 mi/h converted to ft/sec), 'g' is the acceleration due to gravity (32.2 ft/s2) and 'r' is the radius of turn (1000 ft). Next, to find the additional radial force required at 175 mi/h, we use the equation for centripetal force:

F = mv2/r

where 'm' is the mass of the car (3.20 x 10³ lb converted to slugs), 'v' is the speed (175 mi/h converted to ft/sec) and 'r' is the radius of the turn (1000 ft). Subtracting the force provided by the banking angle gives the additional force required.

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