A car whose speed is 90km/hr rounds a curve 180m in radius that is properly banked for a speed of 45km/h. FIind the minimum coefficient of friction between tires and road that wil permit the car to make turn.
I alreadly saw the same question in here. But they uses 'tan'. And in our case we didn't use that. Can anyone help me? :/
To solve this problem, we need to analyze the forces acting on the car as it rounds the curve. The forces involved are the gravitational force (mg), the normal force (N), and the frictional force (f) between the tires and the road.
First, let's find the normal force (N) using the concept of centripetal force. In this case, the centripetal force is provided by the horizontal component of the normal force.
From the given information, we can determine the centripetal force required for the car to make the turn at 90 km/h. The centripetal force (Fc) is given by:
Fc = (mass of the car) × (centripetal acceleration)
The centripetal acceleration is given by:
centripetal acceleration = (speed of the car)^2 / (radius of the curve)
Plugging in the values:
Fc = (mass of the car) × [(90 km/h)^2 / (180 m)]
Now, we can calculate the normal force (N) by taking the horizontal component of the gravitational force (mg):
N = mg × cosθ
In this case, θ is the angle of the banking of the curve (which is not known). We can approximate tanθ by using tanθ = (speed^2) / (g × radius). In this case, we don't know the value of θ, so we need to find it.
Rearranging the equation, we have:
θ = arctan((speed^2) / (g × radius))
Using this approximation for θ, we can calculate the normal force (N):
N = mg × cos(arctan((speed^2) / (g × radius)))
Now, we can determine the minimum coefficient of friction (μ) required for the car to make the turn:
μ = (frictional force) / (normal force)
From the equation for frictional force (f = μN), we can substitute the expression for the normal force and calculate the minimum coefficient of friction:
μ = f / (mg × cos(arctan((speed^2) / (g × radius))))
Simplifying the equation and plugging in the given values will give you the minimum coefficient of friction required for the car to make the turn.