A banked circular highway curve is designed for traffic moving at 57 km/h. The radius of the curve is 211 m. Traffic is moving along the highway at 48 km/h on a rainy day. What is the minimum coefficient of friction between tires and road that will allow cars to negotiate the turn without sliding off the road.
First, convert km/h speeds to m/s.
57 km/h = 15.83 m/s
48 km/h = 13.33 m/s
Next, compute the bank angle of the road, knowing that it is designed for 57 km/h. That means the normal force of the road on the tires is resolvable into a vertical force Mg and a horiziontal force MV^2/R, with no friction force required.
M V^2/R = M g tan A
tan A = V^2/(Rg) = 0.1212
A = 6.9 degrees
Finally, at the lower speed, require that there be a tire friction force Ff sufficient to keep the car from sliding toward the center of the circle.
-Ff + Mg sin A = (M V^2/R)cos A
-M*g*cosA*Us + M*g sin A = (M V^2/R)cos A
Us = -[(V^2/Rg)cos A - g sin A]/(g cosA)
= -V^2/(Rg) + tan A
= -0.0859 + 0.1212 = 0.035
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To determine the minimum coefficient of friction required for cars to negotiate the turn without sliding off the road, we can start by calculating the net force acting on the car as it turns.
The net force acting on the car in a banked curve is the difference between the centripetal force and the gravitational force:
Net Force = Centripetal Force - Gravitational Force
The centripetal force can be calculated using the formula:
Centripetal Force = (mass of the car) x (centripetal acceleration)
The centripetal acceleration can be calculated using the formula:
Centripetal Acceleration = (velocity of the car)^2 / (radius of the curve)
First, let's convert the given velocities from km/h to m/s:
Speed on the curve = 57 km/h = 57 * (1000/3600) = 15.833 m/s
Speed of traffic = 48 km/h = 48 * (1000/3600) = 13.333 m/s
Now, let's calculate the centripetal acceleration:
Centripetal Acceleration = (13.333 m/s)^2 / (211 m)
Centripetal Acceleration = 0.843 m/s^2
Assuming the mass of the car cancels out during calculations, we can simplify the equation to:
Net Force = Centripetal Force - Gravitational Force
Next, let's calculate the gravitational force acting on the car:
Gravitational Force = (mass of the car) x (acceleration due to gravity)
For simplicity, we can use a car mass of 1 kg as the mass eliminates when calculating the coefficient of friction. The acceleration due to gravity is 9.8 m/s^2.
Gravitational Force = (1 kg) x (9.8 m/s^2)
Gravitational Force = 9.8 N
Now, let's calculate the minimum coefficient of friction required:
Net Force = Centripetal Force - Gravitational Force
Since the car is not sliding off the road, the net force is zero:
0 = Centripetal Force - 9.8 N
Rearranging the equation, we get:
Centripetal Force = 9.8 N
Since the force of friction provides the centripetal force, we can use the frictional force formula:
Frictional Force = (coefficient of friction) x (normal force)
In the case of a banked curve, the normal force can be calculated as:
Normal Force = (mass of the car) x (gravitational acceleration)
Since the acceleration due to gravity is 9.8 m/s^2, the normal force is:
Normal Force = (1 kg) x (9.8 m/s^2)
Normal Force = 9.8 N
Now, let's substitute the values into the equation for the frictional force:
Frictional Force = (coefficient of friction) x (normal force)
9.8 N = (coefficient of friction) x (9.8 N)
Simplifying, we find that the minimum coefficient of friction required for the cars to negotiate the turn without sliding off the road is:
Coefficient of friction = 1
Therefore, the minimum coefficient of friction required is 1.