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Hello,

Can you please determine the banking angle needed to make a high speed turn off of a highway exit ramp with a radius of 720 m at a speed of 80 km/h, assuming there is no friction.

I arrived at an answer of: 86 degrees above the horizontal, but that does not seem right.

Thank you.

As the car goes around the banked curve the relevant forces are:

Ff - Force of friction is zero

Fn - perpendicular to and directed away from the road's surface

Fg - the gravitational force straight down

The acceleration will be directed towards the center of the curve - in this case I will assume the this is directed towards the left.

Breaking the normal force into horizontal and vertical components:

Fnv=Fn*cos(alpha) - the vertical component of the normal force

FNh=Fn*sin(alpha) - the horizontal component of the normal force

In the vertical direction there is no acceleration and so opposite force must be equal. In this case that makes the vertical component of the normal force upward equal to the gravitational force downward:

Fg=Fnv becomes m*g=Fn*cos(alpha) [1]

In the horizontal direction the sum of the forces is equal to the mass multiplied by the acceleration. In this case the sum of the forces is the horizontal component of the normal force and the acceleration is centripetal. Newton's 2nd Law becomes:

SF=m*a which becomes Fnh=m*a then Fn*sin(alpha)=m*v^2/R [2] since the acceleration is centripetal.

dividing equation [2] by equation [1]:

Fn*sin(alpha)=m*v^2/R

_____________________

Fn*cos(alpha)=m*g

Simplifying: tan(alpha)=v^2/(R*g)

Therefore, the angle alpha becomes: alpha=atan(v^2/(R*g))

Where: v=80km/hr=80000m/3600s=22.2m/s

Solving for the angle alpha: alpha=atan(22.2^2/(720*9.8)=atan(0.0698)=40 degrees

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I am teaching or have taught AP physics B and C [calculus based mechanics & electricity and magnetism] as well as Lab Physics for college bound students. I have a BS in Physics from the University of Pittsburgh and a Master of Arts in Teaching from same. I have been teaching physics for 34 years. I am constantly updating my skills and have a particular interest in modern physics topics.