What is the smallest radius of an unbanked (flat) track around which a bicyclist can travel if her speed is $35 \mathrm{km} {\mathrm{h}}^{-1}$ and the ${\mu }_s$ between tires and track is $0.40$?

In the figure shown below, a car is driven at constant speed over a circular hill and then into a circular valley with the same radius. At the top of the hill, the normal force on the driver from the car seat is $0$. The driver's mass is $80.0 \mathrm{kg}$. What is the magnitude of the normal force on the driver from the seat when the car passes through the bottom of the valley?

An $85.0 \mathrm{kg}$ passenger is made to move along a circular path of radius $r=3.50 \mathrm{m}$ in uniform circular motion. (a) Figure (a) is a plot of the required magnitude $F$ of the net centripetal force for a range of possible values of the passenger's speed $v$. What is the plot's slope at $v=8.30 \mathrm{m} {\mathrm{s}}^{-1}$? (b) Figure (b) is a plot of $F$ for a range of possible values of $T$, the period of the motion. What is the plot's slope at $T=2.50 \mathrm{s}$? 

An airplane is flying in a horizontal circle at a speed of $600 \mathrm{km} {\mathrm{h}}^{-1}$ (shown in figure). If its wings are tilted at angle $\text{θ}=40°$ to the horizontal, what is the radius of the circle in which the plane is flying? Assume that, the required force is provided entirely by an "aerodynamic lift" that is perpendicular to the wing surface.