Let the radius of the circular segment be

108 m, the mass of the car 1861 kg, and
the coefficient of the static friction between
the road and the tire 0.9. The banking angle
is not given.
Find the magnitude of the normal force N which the road exerts on the car at the opti-mal speed (the speed at which the frictional
force is zero) of 72 km/h.
Answer in units of N.

FBD: friction down angled road/ramp, normal force perpendicular to ramp, gravity down, impending motion up ramp.

Fnet = ma
In r direction,
Friction_r+N_r=mv^2/r
(f_s)(cosX)+NsinX=mv^2/r
uNcosX+NsinX=mv^2/r
N=(mv^2)/(ucosX+sinX) (1)

In z direction, NcosX-uNsinX-W=0
N=mg/(cosX-usinX) (2)

Set equations 1 and 2 equal, solve for X, substitute back into either 1 or 2, yielding N

lol, Equation 1 is actually

N=(mv^2)/(r)(ucosX+sinX)

To find the magnitude of the normal force (N) that the road exerts on the car, we need to consider the forces acting on the car when it is negotiating the circular segment.

We can start by taking a look at the forces acting on the car when it is traveling at the optimal speed (72 km/h).

1. Weight (W): The weight of the car acting vertically downwards. The magnitude of the weight can be calculated using the formula W = mg, where m is the mass of the car (1861 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²).
W = (1861 kg) * (9.8 m/s²)

2. Normal force (N): The normal force is the force exerted by the road perpendicular to the surface. It acts in the upward direction and counters the weight of the car. At the optimal speed, when the frictional force is zero, the normal force and weight are equal in magnitude.
N = W

3. Frictional force (f): The frictional force is responsible for providing the necessary centripetal force that allows the car to move in a circular path. However, at the optimal speed, the frictional force is zero. This means that the car is not relying on friction for its centripetal force, but rather the banking angle of the circular segment.

Given that the coefficient of static friction between the road and the tire is 0.9, we can determine the banking angle needed for the car to negotiate the circular segment.

The formula for the banking angle (θ) is given by:
tan(θ) = v² / (g * r)

Where v is the velocity of the car, g is the acceleration due to gravity, and r is the radius of the circular segment.

In this case, the velocity (v) is 72 km/h, which needs to be converted to m/s:
v = 72 km/h * (1000 m/km) / (3600 s/h)

Given the radius of the circular segment is 108 m, we can solve for the banking angle (θ) using the formula above.

Once we have the banking angle, we can find the normal force (N) by using trigonometry:

N = W * cos(θ)

Calculating all the values and plugging them into the formulas will give us the magnitude of the normal force (N) in units of Newtons.