An 805-kg race car can drive around an unbanked turn at a maximum speed of 54 m/s without slipping. the turn has a radius of curvature of 155 m. air flowing over the car's wing exerts a downward-pointing force (called the downforce) of 10600 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?

This is about relationship between centripetal force and frictional force.

A) Coefficient of static friction = 0.312

B) Max speed with no downward force = 14.16 m/s

We are given;

Mass of car; m = 80 kg

Max speed; v = 54 m/s

Radius of curvature; R = 155 m

Downward force; D = 10600 N

A) We know that formula for centripetal force is;

F = mv²/R

Where;. v is velocity

m is mass

R is radius of curvature

Now, since there is a downward force, the normal force will be;

N = mg + D

Thus, frictional force is given by the formula; F = μN = μ(mg + D)

Where μ is coefficient of static friction

This frictional force will be equal to the centripetal force.

Thus;

μ(mg + D) = mv²/R

Making μ the subject, we have;

μ = mv²/(R(mg + D))

Plugging in the relevant values, we have;

μ = (80 × 54²)/(155((80 × 9.8) + 10600))

μ = 0.132

B) If there is no downward force, then it means D = 0.

Thus;

N = mg + 0

N = mg

Thus;

μmg = mv²/R

m will cancel out to give;

μg = v²/R

v² = μgR

v² = 0.132 × 9.8 × 155

v² = 200.508

v = √200.508

v = 14.16 m/s

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(a) Equating centripetal force to friction force, one finds the relation
  v² = kar
for car speed v, coefficient of friction k, radius of curvature r, and downward acceleration a.

There is already downward acceleration due to gravity. The additional accceleration due to the wing is
  a = F/m = 10600 N/(805 kg) ≈ 13.1677 m/s²
We presume this is added to the 9.80 m/s² gravity provides, so the coefficient of friction is
  k = v²/(ar) = (54 m/s)²/((13.1677 m/s² +9.80 m/s²)·(155 m))
  k ≈ 0.8191

(b) The maximum speed is proportional to the square root of the downward acceleration. Changing that by a factor of 9.80/(9.80+13.17) changes the maximum speed by the square root of this factor.
  max speed with no wing effect = (54 m/s)√(9.8/22.97) ≈ 35.27 m/s

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