A circular curve of radius R in a new highway is designed so that a car travelin

Question

A circular curve of radius R in a new highway is designed so that a car traveling at speed v0 can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from vmin to vmax.

Derive formula for vmax, as a function of µ (coefficient of static friction), v0, and R

Explanation / Answer

In the case in which friction plays no role in holding the car in place gives the speed which is often called the “ideal speed”, or v(ideal). When the speed is greater than v(ideal), then the friction force is relied upon to hold the car in place. The tendency of the car at high speed is to slide up the slope (away from the center of the curve) so friction must act downward. Since the car travels on a curve, a centripetal acceleration is caused. We also have the normal force and the car’s weight (mg) acting on the car (vertically only for mg, of course). The components of the normal force are Nsin? (horizontally) and Ncos? (vertically). The friction force horizontally is µNcos?, and vertically it is µNsin?. So, we can write an equation for the net centripetal forces acting on the car (horizontally): SFx = mv(max)²/R = Nsin? + µNcos? = mv(max)²/R = N(sin? + µcos?)------------------->(1) The net vertical forces sum to zero and allow us to find an expression for the normal force: SFy = 0 = Ncos? - µNsin? - mg N = mg / (cos? - µsin?)---------------------->(2) Now substituting (2) into (1), we can find vmax: mv(max)²/R = [mg / (cos? - µsin?)](sin? + µcos?) v(max)² = gR(sin? + µcos?) /( cos? - µsin?) v(max) = v[ gR(sin? + µcos? )/( cos? - µsin?)] This is sometimes changed to the form of: v(max) = vgR(tan? - µ) / (1 + µtan?)] If you need to see how this form is arrived at, let me know. Hope this helps.

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