A velodrome is built for use in the Olympics (see the figure below). The radius of curvature of the surface is 17.1 m. At what angle should the surface be banked for cyclists moving at 11 m/s? (Choose an angle so that no frictional force is needed to keep the cyclists in their circular path. Large banking angles are used in velodromes.)
To determine the angle at which the surface of the velodrome should be banked, we need to consider the forces acting on the cyclists moving at a constant speed of 11 m/s along the curved path.
First, let's identify the forces involved:
1. Gravity (Weight): The weight of the cyclist and the bicycle acts vertically downward.
2. Normal force: This force acts perpendicular to the surface of the velodrome and prevents the cyclist from sinking into the track. It is equal to the weight of the cyclist and the bicycle.
3. Frictional force: This force acts horizontally, parallel to the surface. In this case, we want to choose an angle that allows us to eliminate the need for a frictional force to keep the cyclist on the curved path.
When no frictional force is required, the only horizontal force acting on the cyclist is the component of the normal force tangential to the circle's curve. This force provides the centripetal force required to keep the cyclist moving in a circular path.
To calculate the vertical component of the normal force, we can use the following equation:
Weight = Normal force × cos(θ)
where θ is the angle of banking.
The centripetal force required at this specific speed can be calculated using the formula:
Centripetal force = Mass × Velocity² / Radius
Now, the centripetal force is also equal to the horizontal component of the normal force:
Centripetal force = Normal force × sin(θ)
By setting these two equations equal and substituting the known values (weight, velocity, radius), we can solve for the angle θ:
Weight × sin(θ) = Mass × Velocity² / Radius
Given that weight = mass × gravity, the equation becomes:
Mass × gravity × sin(θ) = Mass × Velocity² / Radius
Mass cancels out, giving us:
gravity × sin(θ) = Velocity² / Radius
Rearranging the equation, we have:
sin(θ) = Velocity² / (gravity × Radius)
Finally, solving for θ:
θ = arcsin(Velocity² / (gravity × Radius))
Plugging in the given values of Velocity = 11 m/s and Radius = 17.1 m, and using the approximate value for gravity as 9.8 m/s², we can calculate:
θ ≈ arcsin(11² / (9.8 × 17.1))
Evaluating this expression gives us:
θ ≈ arcsin(1.246)
Using a calculator, we find:
θ ≈ 52.8 degrees
Therefore, the surface of the velodrome should be banked at an angle of approximately 52.8 degrees to eliminate the need for a frictional force to keep the cyclists in their circular path.