A velodrome is built for use in the Olympics (see the figure below). The radius of curvature of the surface is 15.9 m. At what angel should the surface be banked for cyclists moving at 15.0 m/s? Choose an angle so that no frictional force is needed to keep the cyclists in their circular path. (Large banking angles are indeed used in velodromes.)

To determine the angle at which the surface of the velodrome should be banked, we can use the concept of a banking angle. The banking angle is the angle at which the surface of the track is tilted with respect to the horizontal plane.

In order for no frictional force to be required to keep the cyclists in their circular path, we need to ensure that the gravitational force along the banking angle provides the necessary centripetal force to keep the cyclists moving in a circular path.

The centripetal force is given by the equation:

Fc = m * v^2 / r

where Fc is the centripetal force, m is the mass of the cyclist, v is the velocity, and r is the radius of curvature of the track.

In this case, we are given the velocity of the cyclists (15.0 m/s) and the radius of curvature (15.9 m). We can assume the mass of the cyclist cancels out in this calculation since it is not given and it does not affect the angle.

Next, let's calculate the gravitational force acting on the cyclist at the banking angle. The gravitational force is given by:

Fg = m * g * cos(θ)

where Fg is the gravitational force, m is the mass of the cyclist, g is the acceleration due to gravity (9.8 m/s^2), and θ is the banking angle.

Since we want the gravitational force to provide the necessary centripetal force, we can equate these two forces:

m * g * cos(θ) = m * v^2 / r

The mass cancels out, and we can solve for the banking angle θ:

g * cos(θ) = v^2 / r
cos(θ) = v^2 / (g * r)
θ = arccos(v^2 / (g * r))

Plugging in the given values, we can calculate the banking angle:

θ = arccos(15.0^2 / (9.8 * 15.9))

θ ≈ 33.5 degrees

Therefore, the surface of the velodrome should be banked at an angle of approximately 33.5 degrees to allow cyclists moving at 15.0 m/s to navigate the track without the need for a frictional force.