Calculating Safe Speed Range on a Banked Curve: Physics Question

[botab]

Hello PF, I need help with a physics question.


A curve of radius 61 is banked for a design speed of 89 .

If the coefficient of static friction is 0.32 (wet pavement), at what range of speeds can a car safely make the curve? [Hint: Consider the direction of the friction force when the car goes too slow or too fast.]

A curve of radius 61 is banked for a design speed of 89 .

How would you find its range? I guess we could find a velocity, but it wouldn't be the the final velocity or the initial velocity?

 

[LowlyPion]

First start by drawing a force diagram of the car on the incline.

You have gravity acting down along the vertical.
You have centrifugal force acting outwardly and horizontal.

You need to resolve these forces into their components relative to the incline, perpendicular and parallel.

Then you have friction acting to retard up the incline motion when it is over 89 and down the incline motion when it is slower.

 

[thomate1]

To calculate the safe speed range on a banked curve, we need to consider the forces acting on the car. The centrifugal force, which pushes the car outwards, is balanced by the normal force from the banked curve and the friction force from the tires. The direction of the friction force depends on the speed of the car.

If the car is going too slow, the friction force will act inwards and help the car stay on the curve. However, if the car is going too fast, the friction force will act outwards and can cause the car to slide off the curve. Therefore, we need to find the speed at which the friction force changes direction from inwards to outwards.

Using the formula for centripetal force, Fc = mv^2/r, we can solve for the speed at which the friction force changes direction. Set the centrifugal force equal to the maximum friction force, which is given by μN, where μ is the coefficient of static friction and N is the normal force.

Fc = mv^2/r = μN

Solving for v, we get v = √(μgr), where g is the acceleration due to gravity. This is the minimum speed at which the car can safely make the curve.

To find the maximum safe speed, we need to consider the maximum friction force, which occurs when the car is at the threshold of sliding off the curve. In this case, the friction force is equal to μN, where N is the normal force acting perpendicular to the banked curve.

Using the same formula for centripetal force, we can solve for the maximum speed, which is given by v = √(μgr + g tanθ), where θ is the angle of the banked curve.

Therefore, the safe speed range for the car on this banked curve is given by √(μgr) ≤ v ≤ √(μgr + g tanθ). Plugging in the given values, we get √(0.32*9.8*61) ≤ v ≤ √(0.32*9.8*61 + 9.8*tanθ), which gives a range of approximately 28.3 m/s to 32.9 m/s.

In conclusion, to find the safe speed range on a banked curve, we need to consider the direction of the friction force and use the formula for