Highway design is strongly influenced by physical analyses, as apparent in const

Question

Highway design is strongly influenced by physical analyses, as apparent in construction of safe circular turns at common highway speeds. The following analyses represent common highway design concerns.

State highway 33 south of Stillwater is scheduled for (extensive) modification between the Cimarron river and US 177 by fall 2012. Dr. Lanee has noticed that one of the curves on this asphalt highway (which is scheduled for replacement) is crowned such that the outside lane is banked away from the turn center rather than into the turn. Consider a car that negotiates this turn, which will be assumed to be symmetrically banked at the road center at angles of 4.00º toward and away from the turn center. For the purposes of these calculations, assume that the turn radius for cars in the inside and outside lanes is identical. What is the radius of this curve for cars on the inside lane to safely negotiate the turn without friction at 65mph (the curve’s posted speed limit)? What is the maximum safe speed for cars in the outside lane without friction? Determine the minimum frictional coefficient for safe travel in the outside lane at the posted speed limit. Is this coefficient reasonable?

Explanation / Answer

The centripetal force will be acting toward the center when the car moves along the banked road.                  N sin ? = MV^2 /R Here N is the normal force, this is the reaction force of the path on the car. the weight is equal to the component force as             N cos ? = mg Dividing the equation sin ? / cos ? = V^2 / Rg the radius of the path is R = V^2 / g tan ?                                              = [(65mi/hr)( 1609m/mi)(1hr/3600)]^2 / (9.8) tan 4                                              = 1231.58m ---------------------------------------------------------------------------------------- The maximum permissble speed is obtained                    V = [ Rg ]^(1/2)                       = [ 1231.58 x 9.8 ]^(1/2)                       = 109.83m/s --------------------------------------------------------------------------------------------- The coeficient of friction for this maximum permissable speed is obtained               µ M g = MV^2 / R               µ = V^2 / R g                  = (109.83)^2 / 1231.83(9.8)                  = 0.99    The coefficent of friction is too high. The centripetal force will be acting toward the center when the car moves along the banked road.                  N sin ? = MV^2 /R Here N is the normal force, this is the reaction force of the path on the car. the weight is equal to the component force as             N cos ? = mg Dividing the equation sin ? / cos ? = V^2 / Rg the radius of the path is R = V^2 / g tan ?                                              = [(65mi/hr)( 1609m/mi)(1hr/3600)]^2 / (9.8) tan 4                                              = 1231.58m ---------------------------------------------------------------------------------------- The maximum permissble speed is obtained                    V = [ Rg ]^(1/2)                       = [ 1231.58 x 9.8 ]^(1/2)                       = 109.83m/s --------------------------------------------------------------------------------------------- The coeficient of friction for this maximum permissable speed is obtained               µ M g = MV^2 / R               µ = V^2 / R g                  = (109.83)^2 / 1231.83(9.8)                  = 0.99    The coefficent of friction is too high. The centripetal force will be acting toward the center when the car moves along the banked road.                  N sin ? = MV^2 /R Here N is the normal force, this is the reaction force of the path on the car. the weight is equal to the component force as             N cos ? = mg Dividing the equation sin ? / cos ? = V^2 / Rg the radius of the path is R = V^2 / g tan ?                                              = [(65mi/hr)( 1609m/mi)(1hr/3600)]^2 / (9.8) tan 4                                              = 1231.58m ---------------------------------------------------------------------------------------- The maximum permissble speed is obtained                    V = [ Rg ]^(1/2)                       = [ 1231.58 x 9.8 ]^(1/2)                       = 109.83m/s --------------------------------------------------------------------------------------------- The coeficient of friction for this maximum permissable speed is obtained               µ M g = MV^2 / R               µ = V^2 / R g                  = (109.83)^2 / 1231.83(9.8)                  = 0.99    The coefficent of friction is too high.

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