If a curve with a radius of 94.5 m is perfectly banked for a car traveling 72.4 km/h, what must be the coefficient of static friction for a car not to skid when traveling at 96.4 km/h?
First compute the "perfect bank" angle at the 72.4 km/h speed.
Then require that the centripetal force at the higher speed be provided by the maximum friction force compatible with the static friction coefficient.
To determine the coefficient of static friction required for a car not to skid when traveling at a certain speed, we need to consider the forces acting on the car at the maximum speed.
First, let's convert the given speeds to meters per second (m/s).
72.4 km/h = (72.4 * 1000) / (60 * 60) m/s ≈ 20.11 m/s
96.4 km/h = (96.4 * 1000) / (60 * 60) m/s ≈ 26.78 m/s
Next, we can analyze the forces acting on the car when it is traveling at the maximum speed of 26.78 m/s.
1. Centripetal Force (Fc):
The centripetal force is the force that keeps an object moving in a curved path and is given by the equation Fc = (mv^2) / r, where m is the mass of the car, v is its velocity, and r is the radius of the curve.
In this case, since we are given the radius (94.5 m), we need to find the mass of the car. Since the mass is canceled out when calculating the coefficient of static friction, we can ignore it and focus only on the velocities and radius.
2. Weight (mg):
The weight of the car creates a downward force, which is given by the equation mg, where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
3. Normal Force (Fn):
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, it acts perpendicular to the surface of the road. When the car is perfectly banked, the normal force is split into two components: the vertical component supports the weight of the car (Fn = mg), and the horizontal component balances the centripetal force.
With these forces in mind, we can write the equation for the horizontal component of the normal force (Fn_horizontal):
Fn_horizontal = mg * tan(θ)
Since the car is perfectly banked, the angle θ can be calculated using the formula:
tan(θ) = v^2 / (r * g)
Plugging in the given values, we have:
tan(θ) = (26.78^2) / (94.5 * 9.8)
θ ≈ 26.95 degrees
Now, we need to consider the forces acting on the car when it is traveling at the lower speed of 20.11 m/s.
Using the formula for the horizontal component of the normal force, we have:
tan(θ) = (20.11^2) / (94.5 * 9.8)
θ ≈ 17.78 degrees
Now, we can calculate the coefficient of static friction (μs) using the difference in angles.
μs = tan(θ_max) - tan(θ_min)
= tan(26.95 degrees) - tan(17.78 degrees)
≈ 0.54
Therefore, the coefficient of static friction required for the car not to skid when traveling at 96.4 km/h is approximately 0.54.