Physics Banked curve multi-part

Question

When you are driving around a banked curve, you will see a recommended speed limit. Many factors go into this determination, and physics plays a role. In this problem, you will work through the analysis to determine the maximum speed you can drive around the curve in the absence of friction (for example if the road is slippery due to the weather).

(a) A front-view of the car is shown below. Draw a free body diagram representing the forces acting on the car.

(b) This is one situation where rotating the coordinate axes to align with the incline is not beneficial. For the coordinate axes shown above (+x right, +y up), determine the angle between the normal force and the y axis. This involves geometry, not physics. It may help to make a separate drawing with the triangle that represents the road, a line extending from the road representing the normal force, and the coordinate axes centered on the intersection between the road and the line representing the normal force.

(c) Apply Newton’s second law to the y components of force and motion to determine the magnitude of the normal force in terms of the mass m of the car and the angle θ the road makes with the horizontal.

(d) Apply Newton’s second law to the x components of force and motion to determine the speed. Your answer will only involve the constant g, the angle θ, and the radius r of the curve

Answer 1

a) Without friction, you have just a vertical weight mg vector and a normal force F_N vector that is perpendicular to the ramp.

b) From the suggested sketch you should see that the angle between F_N and the vertical is the same as angle θ between the road and the horizontal.

c) ΣF_y = F_N cos θ - mg = 0, F_N = mg / cos θ.

d) ΣF_x = -F_N sin θ = ma = -mv² /r, mv² /r = (mg / cos θ) • sin θ, v² = rg tan θ, v = √(rg tan θ).