As you travel every day to campus, the road makes a large turn that is approxima

Question

As you travel every day to campus, the road makes a large turn that is approximately an arc of a circle. You notice the warning sign at the start of the turn, asking for a maximum speed of 55 mi/h. You also notice that in the curved portion the road is level-that is, not banked at all. On a dry day with very little traffic, you enter the turn at a constant speed of 80 mi/h and feel that the car may skid if you do not slow down quickly. You conclude that your speed is at the limit of safety for this curve and you slow down. However, you remember reading that on dry pavement new tires have an average coefficient of static friction of about 0.76, while under the worst winter driving conditions, you may encounter wet ice for which the coefficient of static friction can be as low as 0.20. Wet ice is not unheard of on this road, so you ask yourself whether the speed limit for the turn on the roadside warning sign is for the worst-case scenario. A. Estimate the radius of the curve from your 80 mi/h experience in the dry turn? B. Use this estimate to find the maximum speed limit in the turn under the worst wet-ice conditions?

Explanation / Answer

A) friction force f = uN = umg

friction force = centripetal force

u mg = mv^2 / r

ug = v^2 / r

so u = 0.76 and v = 80 x 1609 m / 3600s =35.76

0.76 x 9.81 = 35.76^2 / r

r = 171.48 m

b) noq u = 0.20

u g = v^2 / r

0.20 x 9.81 = v^2 / 171.48

v = 18.34 m/s or 18.34 x 3600/1609 = 41.04 mi /h

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