A 850-kg race car can drive around an unbanked turn at a maximum speed of 61 m/s without slipping. The turn has a radius of 160 m. Air flowing over the car's wing exerts a downward-pointing force (called the downforce) of 11000 N on the car. (a) What is the coefficient of static friction between the track and the car's tires? (b) What would be the maximum speed if no downforce acted on the car?
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To find the coefficient of static friction between the track and the car's tires, we can use the centripetal force equation:
F_centripetal = m * a_centripetal
where F_centripetal is the centripetal force, m is the mass of the car, and a_centripetal is the centripetal acceleration. The centripetal force is given by the sum of the gravitational force (mg) and the downforce (F_down):
F_centripetal = mg + F_down
Let's calculate these values:
(a) Finding the coefficient of static friction:
1. Calculate the centripetal force (F_centripetal):
F_centripetal = m * a_centripetal
F_centripetal = 850 kg * a_centripetal
2. Calculate the downforce (F_down):
F_down = 11000 N
3. Calculate the gravitational force (mg):
Using the formula F_gravitational = mg, we can calculate mg:
F_gravitational = mg
850 kg * g = m * g
850 kg * 9.8 m/s^2 = m * 9.8 m/s^2
4. Substitute the values into the equation for F_centripetal:
F_centripetal = F_gravitational + F_down
850 kg * a_centripetal = 850 kg * 9.8 m/s^2 + 11000 N
5. Solve for a_centripetal:
a_centripetal = (850 kg * 9.8 m/s^2 + 11000 N) / 850 kg
Next, we can calculate the maximum speed if no downforce acted on the car.
(b) Finding the maximum speed without downforce:
1. In the absence of the downforce, the centripetal force would only be provided by the friction force between the tires and the track. Therefore, F_centripetal = μs * (m * g), where μs is the coefficient of static friction.
2. Substitute the values into the equation for F_centripetal:
μs * (m * g) = (m * a_centripetal)
3. Solve for μs:
μs = (m * a_centripetal) / (m * g)
Now we have the coefficient of static friction (μs) and the maximum speed without downforce.