A curve of radius 72 is banked for a design speed of 120 .If the coefficient of static friction is 0.35 (wet pavement), at what range of speeds can a car safely handle the curve?

To determine the range of speeds at which a car can safely handle the curve, we need to consider the forces acting on the car.

Let's start by calculating the angle of banking of the curve. The angle of banking (θ) can be calculated using the formula:

θ = arctan(v^2 / (g * r))

Where:
- v is the speed of the car
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- r is the radius of the curve

So, in this case:
- v = 120 mph (convert to m/s: 1 mph = 0.44704 m/s, so 120 mph = 53.6448 m/s)
- g = 9.8 m/s^2
- r = 72 m

Plugging these values into the formula, we get:

θ = arctan((53.6448 m/s)^2 / (9.8 m/s^2 * 72 m))
≈ arctan(8.8624)

Using a calculator, we find that θ ≈ 83.2 degrees.

Now, let's consider the forces acting on the car when it is banking. The two main forces are the centripetal force (F_c) and the frictional force (F_f).

The centripetal force can be calculated using the formula:

F_c = m * v^2 / r

Where:
- m is the mass of the car

The frictional force can be calculated using the formula:

F_f = μ * m * g

Where:
- μ is the coefficient of static friction
- m is the mass of the car
- g is the acceleration due to gravity

For the car to safely handle the curve, the frictional force must be equal to or greater than the centripetal force:

F_f ≥ F_c

Substituting the formulas for F_c and F_f, we get:

μ * m * g ≥ m * v^2 / r

Canceling out the mass (m) on both sides of the inequality, we have:

μ * g ≥ v^2 / r

Now, let's plug in the values:

μ = 0.35 (coefficient of static friction)
g = 9.8 m/s^2 (acceleration due to gravity)
v = 120 mph ≈ 53.6448 m/s (speed of the car)
r = 72 m (radius of the curve)

0.35 * 9.8 m/s^2 ≥ (53.6448 m/s)^2 / 72 m

Simplifying, we get:

3.43 ≥ 399.82 / 72

Multiply both sides by 72:

3.43 * 72 ≥ 399.82

246.96 ≥ 399.82

Since this inequality is false, it means that the car cannot safely handle the curve at a speed of 120 mph (or 53.6448 m/s) on wet pavement with the given coefficient of static friction.

To determine the safe range of speeds, we would need to repeat the calculation for different speeds and find the highest speed that satisfies the inequality.