A car rounds a banked curve where the radius of curvature of theroad is R , the

Question

A car rounds a banked curve where the radius of curvature of theroad is R, the banking angle is , and thecoefficient of static friction is µ. (a) Determine the range of speeds the car canhave without slipping up or down the road. (Use mu forµ, theta for , and R andg as appropriate in your equations.)
minimum speed 1 maximum speed 2

(b) What is the range of speeds possible if R =150 m, = 10°, andµ = 0.11 (slipperyconditions)?
minimum speed 3 m/s maximum speed 4 m/s (a) Determine the range of speeds the car canhave without slipping up or down the road. (Use mu forµ, theta for , and R andg as appropriate in your equations.)
minimum speed 1 maximum speed 2

(b) What is the range of speeds possible if R =150 m, = 10°, andµ = 0.11 (slipperyconditions)?
minimum speed 3 m/s maximum speed 4 m/s minimum speed 1 maximum speed 2

Explanation / Answer

(a)The normal force n is perpendicular to the roadway at anangle with the vertical.The acceleration in the x-directionis the centripetal acceleration,there is no acceleration in they-direction.Thus the equations of Newton's second law are Fx = n * sin = m *arad Fy = n * cos + (-mg) = 0 From the Fy equation,n =(mg/cos).Substituting this into the Fxequation gives an expression for the banking angle: tan = (arad/g) Finally,substituting the expression arad =(v2/R),we have tan = (v2/g * R) -----------(1) The minimum speed is obtained when = 0o andthe maximum speed is obtained when = 45o. (b)The minimum speed is obtained when =0o. From equation (1) we get v2 = g * R * tan or v2 = g * R * tan (since =) or v = (g * R * tan)1/2 or v = ( * g * R * tan)1/2 (underslippery conditions) or v = ( * g * R * tan)1/2 (underslippery conditions) The maximum speed is obtained by substituting R =150 m, = 10°, andµ = 0.11

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