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?
To find the coefficient of static friction between the track and the car's tires, we can use the following equation:
F_friction = μ_s * F_normal
where:
F_friction = force of friction between the tires and track
μ_s = coefficient of static friction
F_normal = normal force on the car (equal to the car's weight)
(a) First, let's find the normal force on the car:
F_gravity = m * g
where:
m = mass of the car
g = acceleration due to gravity (9.8 m/s^2)
F_gravity = (850 kg) * (9.8 m/s^2)
F_gravity = 8330 N
The normal force is equal to the sum of the weight and the downward force (downforce) exerted by the air on the car's wing:
F_normal = F_gravity + F_downforce
F_normal = 8330 N + 11000 N
F_normal = 19330 N
Now we can substitute the values into the equation for force of friction:
F_friction = μ_s * F_normal
F_friction = μ_s * 19330 N
We also know that the centripetal force keeping the car in the turn is equal to the force of friction:
F_centripetal = F_friction
F_centripetal = (m * v^2) / r
where:
m = mass of the car
v = velocity of the car
r = radius of the turn
Substituting the given values:
F_centripetal = (850 kg) * (61 m/s)^2 / 160 m
F_centripetal = 19146.875 N
Since F_centripetal = F_friction:
μ_s * 19330 N = 19146.875 N
Now we can solve for the coefficient of static friction, μ_s:
μ_s = 19146.875 N / 19330 N
μ_s ≈ 0.992
Therefore, the coefficient of static friction between the track and the car's tires is approximately 0.992.
(b) To find the maximum speed if no downforce acted on the car, we can use the same equation for the centripetal force:
F_centripetal = (m * v^2) / r
But this time, the force of friction is not affected by the downforce:
F_centripetal = F_friction
F_centripetal = μ_s * F_normal
Substituting the given values:
μ_s * F_normal = (m * v^2) / r
Solving for v:
v^2 = (μ_s * F_normal * r) / m
v = sqrt((μ_s * F_normal * r) / m)
Substituting the given values:
v = sqrt((0.992 * 19330 N * 160 m) / 850 kg)
v ≈ 54.36 m/s
Therefore, the maximum speed if no downforce acted on the car would be approximately 54.36 m/s.