A banked turn in a new highway has been designed so that any vehicle traveling o

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A banked turn in a new highway has been designed so that any vehicle traveling on glare ice at v1=50 km/h will successfully complete the turn at the speed limit. The radius of the turn is r=185 m. For good road conditions, the posted speed limit is v2=80 km/h. What coefficient of static friction must any vehicle have to drive at the speed limit during good road conditions? (I) Derive an algebraic expression for the road’s banking angle , theta = ?
Then calculate the banking angle (II) Derive an algebraic expression for the force of friction during good road conditions, for a car with speed v friction = ? Then calculate the force of friction (III) Derive an algebraic expression for the normal force during good road conditions, for a car with speed v
n = ? Then calculate the normal force (IV) Derive an algebraic expression for the minimum coefficient of static friction during good road conditions, for a car with speed v
Coefficient = ? Then calculate the minimum coefficient of static friction A banked turn in a new highway has been designed so that any vehicle traveling on glare ice at v1=50 km/h will successfully complete the turn at the speed limit. The radius of the turn is r=185 m. For good road conditions, the posted speed limit is v2=80 km/h. What coefficient of static friction must any vehicle have to drive at the speed limit during good road conditions? (I) Derive an algebraic expression for the road’s banking angle , theta = ?
Then calculate the banking angle (II) Derive an algebraic expression for the force of friction during good road conditions, for a car with speed v friction = ? Then calculate the force of friction (III) Derive an algebraic expression for the normal force during good road conditions, for a car with speed v
n = ? Then calculate the normal force (IV) Derive an algebraic expression for the minimum coefficient of static friction during good road conditions, for a car with speed v
Coefficient = ? Then calculate the minimum coefficient of static friction A banked turn in a new highway has been designed so that any vehicle traveling on glare ice at v1=50 km/h will successfully complete the turn at the speed limit. The radius of the turn is r=185 m. For good road conditions, the posted speed limit is v2=80 km/h. What coefficient of static friction must any vehicle have to drive at the speed limit during good road conditions? (I) Derive an algebraic expression for the road’s banking angle , theta = ?
Then calculate the banking angle (II) Derive an algebraic expression for the force of friction during good road conditions, for a car with speed v friction = ? Then calculate the force of friction (III) Derive an algebraic expression for the normal force during good road conditions, for a car with speed v
(II) Derive an algebraic expression for the force of friction during good road conditions, for a car with speed v friction = ? Then calculate the force of friction (III) Derive an algebraic expression for the normal force during good road conditions, for a car with speed v
n = ? Then calculate the normal force (IV) Derive an algebraic expression for the minimum coefficient of static friction during good road conditions, for a car with speed v
Coefficient = ? Then calculate the minimum coefficient of static friction

Explanation / Answer

let then banking angle be theta
and the coefficient of friction be k
given, speed limit for travelling on glare ice, v1 = 50 km/h = 13.889 m/s
speed limit for good road conditions, v2 = 80 km/h = 22.22 m/s

i) for glare ice, k = 0
   then from force balance
   let N be the normal force of reaction between the road and the car
   then Ncos(theta) = mg ( where m is mass of the car)
   Nsin(theta) = m(v1)^2/R
   where R = 185 m ( given )
   hence
   tan(THETA) = (v1)^2/R*g = (13.889^2/185*9.81)
   theta = 6.607 deg

ii) now, for good road conditions, let force of friction be f
   then
   f = mv^2*cos(theta)/R - mgsin(theta) [ for a car moving with speed v]
iii) N = mgcos(theta) + mv^2*sin(theta)/R [ for a car moving with speed v]
iv) now, f = kN
   f/N = k =    [mv^2*cos(theta)/R - mgsin(theta)]/[mgcos(theta) + mv^2*sin(theta)/R]
   f/N = k =    [v^2*cos(theta)/R - gsin(theta)]/[gcos(theta) + v^2*sin(theta)/R]
   for v = 22.22 m/s
   f/N = [22.22^2*cos(6.607)/185 - 9.81*sin(6.607)]/[9.81*cos(6.607) + 22.22^2*sin(6.607)/185]
   f/N = k = 0.02287
   so minimum coefficient of static friction = 0.02287

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