Circular highway; minimize radius under certain parameters

[khfrekek92]

Homework Statement

These conditions include the following:
(1)radius must be as small as possible.
(2)It needs to be possible to park a car on the curve without sliding off on an icy day when coefficients of static and kinetic friction are .2 and .1 respectively.
(3)Cars need to make the curve at 75 mph even if there were no friction.

What is this radius, and what angle should the highway be banked above the horizontal?

Homework Equations

f=ma, F_f=uN

The Attempt at a Solution

I've set up the force-body diagrams with gravity, normal, and friction. Next, I know I need to use F=ma, right? I am lost on how I set this up to minimize this radius and figure out what the bank would need to be.. Any help? Thanks so much in advance!

 

[anathema]

I would approach this problem by first determining the necessary centripetal force for a car to make the curve at 75 mph. This can be calculated using the formula F=mv^2/r, where m is the mass of the car, v is the velocity, and r is the radius of the curve.

Next, I would consider the forces acting on the car as it makes the curve. The normal force (N) and the frictional force (F_f) must balance the centripetal force in order for the car to stay on the curve without sliding off. The maximum frictional force can be calculated using the formula F_f=uN, where u is the coefficient of static friction.

To ensure that the car can make the curve even without friction, we need to consider the minimum centripetal force required at a speed of 75 mph. This can be calculated by setting the frictional force to zero in the formula F=mv^2/r.

Now, to minimize the radius of the curve, we need to minimize the centripetal force required. This can be achieved by increasing the velocity of the car, as the mass and radius are fixed. However, we also need to consider the maximum frictional force that the car can handle without sliding off. Therefore, we need to find the maximum velocity at which the car can make the curve without exceeding the maximum frictional force. This can be calculated by setting the frictional force equal to the maximum frictional force and solving for the velocity.

Once we have determined the necessary velocity, we can use the formula F=mv^2/r to calculate the minimum radius for the curve.

To determine the angle of banking for the highway, we can use the formula tanθ=v^2/(rg), where θ is the angle of banking, v is the velocity, r is the radius, and g is the acceleration due to gravity. This will give us the angle above the horizontal at which the highway should be banked to ensure that the car can make the curve at 75 mph without sliding off.

In summary, as a scientist, I would approach this problem by first considering the necessary centripetal force and the maximum frictional force for the car to make the curve at 75 mph. Then, I would use these values to determine the minimum radius for the curve and the angle of banking for the highway.