Determine the vehicle speed for which this curve is appropriate

[pezzang]

Here is the question:

Q. A highway curve that has a radius of curvature of 100m is banked at an angle of 15 degree.

a) Determine the vehicle speed for which this curve is appropriate if there is no friction between the road and the tires of the vehicle.


On a dry day when friction is present, an automobile successfully negotiates the curve at a speed of 25m per second.
b) Tell me all the forces on the automobile.
c) Determine the minimum value of the coefficient of friction necessary to keep this automobile from sliding as it goes around the curve.

I would appreciate your response to each question!

 

[HallsofIvy]

And we would appreciate seeing what you have tried so far!
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[M98Ranger]

a) The vehicle speed for which this curve is appropriate can be determined using the formula v = √(rgtanθ), where v is the speed, r is the radius of curvature, g is the gravitational acceleration (9.8m/s^2), and θ is the angle of banking. Plugging in the given values, we get v = √(100*9.8*tan15) = 21.9m/s. Therefore, the appropriate speed for this curve is approximately 21.9m/s or 78.8km/h.

b) The forces acting on the automobile are the normal force (N), the weight of the car (mg), and the centripetal force (Fc). The normal force is perpendicular to the surface of the road and balances the weight of the car. The weight of the car is acting downwards due to gravity. The centripetal force is directed towards the center of the curve and is responsible for keeping the car in a circular motion.

c) In order to keep the car from sliding as it goes around the curve, the centripetal force must be equal to or greater than the maximum static friction force. The maximum static friction force can be calculated using the formula Fmax = μN, where μ is the coefficient of friction and N is the normal force. Plugging in the given values, we get Fmax = μmg. Setting this equal to the centripetal force (Fc = mv^2/r), we can solve for the minimum value of μ. Therefore, μ = Fc/mg = (mv^2)/mgr = v^2/gr = (25^2)/(100*9.8) = 0.64. Therefore, the minimum value of the coefficient of friction necessary to keep the car from sliding is 0.64.