Circular Motion on a Bank Question

[avonrepus]

I have a general question about circular motion of a car on a
frictionless bank.

What would be the function of the path (what is the shape?)
of the car entering a bank with a velocity v, and is slipping upwards
because the v is too high to for mv^2/r = Nsinθ.

The initial velocity is going straight into the page (When the cross section view of the bank is made)
The path is to be studied from when it enters the bank to where it is moving
in circular motion.

Please describe the steps needed to solve this problem.

 

[Nate810]

The path of the car is determined by three main factors: the initial velocity, the radius of curvature, and the coefficient of friction. Since the bank is assumed to be frictionless, the coefficient of friction is zero and the car will travel in a straight line until it reaches the point where the centripetal acceleration (i.e. mv^2/r) equals the normal force (Nsinθ). At this point, the car's path will bend sharply and begin to follow a circular path around the bank. To solve for the path more precisely, we can use the equation of motion for uniform circular motion, which states that the radial acceleration (ar), the tangential acceleration (at), and the angular velocity (ω) are related by the equation ar = ω^2r + at. The angular velocity can be calculated from the initial velocity and the radius of curvature, and then the radial and tangential accelerations can be calculated from the angular velocity. Finally, the equation of motion can be solved to determine the path of the car.

 

[m2003]

The shape of the path of the car on a frictionless bank would be a curve, specifically a section of a circle. This is because the car's motion is constrained by the bank and the forces acting on it, causing it to move in a circular path.

To solve this problem, we first need to understand the forces acting on the car. The car has two main forces acting on it: the normal force (N) from the bank and the centripetal force (Fc) towards the center of the circular path. The normal force is perpendicular to the bank and prevents the car from slipping off, while the centripetal force is responsible for keeping the car moving in a circular path.

Next, we need to find the relationship between the velocity (v) of the car and the radius (r) of the circular path. This can be done by equating the centripetal force (Fc = m*v^2/r) to the normal force (N = mg*cosθ) multiplied by the sine of the angle of the bank (sinθ). Solving for r, we get r = v^2/(g*cosθ*tanθ).

Now, we can plot the path of the car by using this equation and varying the values of v and θ. For a given velocity v, as the angle of the bank θ increases, the radius of the circular path decreases, resulting in a sharper curve.

To find the exact shape of the path, we can use the equation for a circle (x^2 + y^2 = r^2) and substitute the value of r from the previous equation. This will give us the equation for the path of the car.

In summary, to solve this problem, we need to understand the forces acting on the car, find the relationship between velocity and radius, and use this to plot the path and determine its shape.