Problem:
A cyclist is riding a bicycle at a speed of 14(3)^1/2 m/s and takes a turn around a circular road of radius 20(3)^1/2 m without skidding. What is his inclination to the vertical?

Solution:

Understanding the Problem:
In this problem, we need to find the inclination of the cyclist to the vertical while taking a turn around a circular road. The cyclist is moving at a constant speed and is not skidding, which means the net force acting on the cyclist is towards the center of the circular path.

Key Information:
- Speed of the cyclist: 14(3)^1/2 m/s
- Radius of the circular road: 20(3)^1/2 m

Analysis:
To solve this problem, we will use the concept of centripetal force. The centripetal force is the force that keeps an object moving in a circular path and is always directed towards the center of the circle. In this case, the centripetal force is provided by the friction between the bicycle tires and the road.

Calculating the Centripetal Force:
The centripetal force can be calculated using the formula:

F = (m * v^2) / r

Where:
F = Centripetal force
m = Mass of the cyclist
v = Velocity of the cyclist
r = Radius of the circular path

In this problem, we are not given the mass of the cyclist, but we can assume it to be negligible compared to the other forces involved. Therefore, we can ignore the mass and calculate the centripetal force using the given values.

Finding the Inclination:
The inclination of the cyclist to the vertical can be found using the concept of tangent. In a circular path, the tangent is perpendicular to the radius at the point where the cyclist is located.

Calculating the Inclination:
The inclination can be calculated using the formula:

θ = tan^(-1) (v^2 / (g * r))

Where:
θ = Inclination to the vertical
v = Velocity of the cyclist
g = Acceleration due to gravity
r = Radius of the circular path

In this problem, we are not given the value of the acceleration due to gravity, but we can assume it to be approximately 9.8 m/s^2.

Final Answer:
By substituting the given values into the formula, we can calculate the inclination of the cyclist to the vertical.