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Chapter 9: Problem 43

An automobile engine has a maximum power output of \(70 \mathrm{hp},\) which occurs at an engine speed of \(2200 \mathrm{rpm} . \mathrm{A} 10 \%\) power loss occurs through the transmission and differential. The rear wheels have a radius of 15.0 in. and the automobile is to have a maximum speed of 75 mph along a level road. At this speed, the power absorbed by the tires because of their continuous deformation is 27 hp. Find the maximum permissible drag coefficient for the automobile at this speed. The car's frontal area is \(24.0 \mathrm{ft}^{2}\)

Short Answer

Expert verified
The maximum permissible drag coefficient for the automobile at the speed of 75 mph is approximately 0.311 (Cd = 0.311) when rounded to three decimal places.

Step by step solution

01

Conversion of Units

First, convert all the given quantities to SI units. Convert the maximum power output from horsepower to watts (since 1 hp = 746 watts). Convert the radius of the wheels from inches to metres (since 1 inch = 0.0254 meters). Convert the speed from miles per hour to metres per second (since 1 mph = 0.44704 m/s). Convert the frontal area from square feet to square metres (since 1 ft² = 0.092903 m²).
02

Calculate Available Power at Wheels

Afterwards, calculate the power available at the wheels. The 10% power loss through the transmission and differential translates to a 90% transmission of power to the wheels. Take 90% of the engine's maximum power to achieve this.
03

Relate Power to Speed

To relate the power available at the wheels to the speed, we need to subtract the power absorbed by the tires from the power available at the wheels. From the remaining power, we can calculate the force exerted by the tires using the power formula \(P = F * v\), where \(P\) is the power, \(F\) is the force, and \(v\) is the speed. Solve this formula for \(F\).
04

Calculate the Drag Coefficient

Lastly, calculate the drag coefficient. This can be done by setting the force from Step 3 equal to the drag force equation, \(F = 0.5 * Cd * p * A * v^2\), where \(Cd\) is the drag coefficient, \(p\) is the air density (approx. 1.23 kg/m³), \(A\) is the car's frontal area, and \(v\) is the speed. Solve this formula for \(Cd\).

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