In the Daytona 500 car race in Daytona, FL, a curve in the oval track has a radius of 316 m (near

the top of the curve) and is banked at 31.0
O
.
a) What speed would be necessary to make this turn if there was no friction on the road?
b) If the coefficient of static friction between the tires and the road is 0.70, what are the minimum
and maximum speeds that a car could take this turn and not slide?

(a)

x: ma=N•sinα
y: 0=N•cosα –mg, => N=mg/cosα
a=v²/R
m• v²/R = N•sinα= mg•sinα/cosα= mg•tanα
v=sqrt(R• g•tanα)
(b)
x: ma=N•sinα+F(fr) •cosα
y: mg= N•cosα –mg-F(fr) •sinα

Since F(fr)=μ•N,
ma=N(sinα+μ•cosα)
mg= N(cosα - μ• sinα)
ma/mg = (sinα+μ•cosα)/ (cosα - μ• sinα)
a=v² /R
v² /Rg= (sinα+μ•cosα)/ (cosα - μ• sinα)
v=sqrt[Rg (sinα+μ•cosα)/ (cosα - μ• sinα)]

To answer these questions, we need to apply the concepts of centripetal force and friction. The centripetal force is the force that keeps an object moving along a curved path. In this case, it is provided by the horizontal component of the normal force acting on the car due to the banking of the track.

We can solve for the speed required using the centripetal force equation:

(a) No friction:

The centripetal force is provided solely by the horizontal component of the normal force.

F_centripetal = m * (v^2 / r)

Where:
F_centripetal is the centripetal force
m is the mass of the car
v is the velocity of the car
r is the radius of the curve

Since there is no friction, the normal force N is equal to the car's weight, which is given by:

N = m * g

Where g is the acceleration due to gravity.

The horizontal component of N, which provides the centripetal force, is given by:

N_horizontal = N * cosθ

Where θ is the angle of the banking.

Set N_horizontal equal to the centripetal force:

N_horizontal = F_centripetal
m * g * cosθ = m * (v^2 / r)

Cancel out the mass m:

g * cosθ = v^2 / r

Solve for v:

v = sqrt(g * r * cosθ)

Substituting the given values:
g = 9.8 m/s^2 (acceleration due to gravity)
r = 316 m (radius)
θ = 31.0° (angle of banking)

v = sqrt(9.8 * 316 * cos(31.0°))

Calculate this expression to find the speed required to make this turn with no friction.

(b) With friction:

The maximum and minimum speeds can be found using the coefficient of static friction, μ.

The maximum speed occurs when the static friction reaches its maximum value. The formula for the maximum speed is:

v_max = sqrt(μ * g * r * cosθ)

The minimum speed occurs when the static friction reaches its minimum value. The formula for the minimum speed is:

v_min = sqrt(μ * g * r * sinθ)

Substituting the given values:
μ = 0.70 (coefficient of static friction)
g = 9.8 m/s^2 (acceleration due to gravity)
r = 316 m (radius)
θ = 31.0° (angle of banking)

Calculate these expressions to find the minimum and maximum speeds that a car could take this turn and not slide.