A curve along the highway is to be designed for cars moving at 100km/hr. Find the angle of banking if the radius of the curve is 200m
Ac = v^2/R
mg sin a = mv^2/R
so
sin a = v^2/(R g)
and
v = 100,000/ 3,600 = 27.8 m/s
solve for angle a
To find the angle of banking, we can consider the forces acting on the car as it moves along the curve. The three main forces involved are the gravitational force, the normal force, and the friction force.
1. Gravitational Force (Fg): This force acts vertically downwards, and its value is given by the equation Fg = m * g, where m is the mass of the car and g is the acceleration due to gravity.
2. Normal Force (Fn): This force acts perpendicular to the curve and counterbalances the gravitational force. It prevents the car from sinking into the road or flying off the curve. Its value can be calculated using the equation Fn = m * v^2 / r, where m is the mass of the car, v is the velocity of the car, and r is the radius of the curve.
3. Friction Force (Ff): This force acts horizontally towards the center of the curve and provides the necessary centripetal force to keep the car moving in a circular path. The friction force is given by Ff = µ * Fn, where µ is the coefficient of static friction between the tires of the car and the road.
Since the car is moving at a constant speed, we know that the gravitational force and the friction force are equal in magnitude. Therefore, we can equate Fg and Ff to find the coefficient of static friction (µ). The equation becomes:
m * g = µ * m * v^2 / r
We can cancel out the mass (m) from both sides of the equation:
g = µ * v^2 / r
Now, we can rearrange the equation to solve for µ:
µ = (g * r) / v^2
Finally, we can calculate the angle of banking (θ) using the equation:
θ = atan(µ)
Let's plug in the given values:
Radius of the curve (r) = 200m
Velocity of the car (v) = 100km/hr = 100,000m/3,600s ≈ 27.78m/s
Acceleration due to gravity (g) = 9.8m/s^2
Now, substituting these values into the equation for µ, we can solve for µ:
µ = (9.8m/s^2 * 200m) / (27.78m/s)^2 ≈ 0.413
Finally, we can calculate the angle of banking:
θ = atan(0.413) ≈ 22.7 degrees
Therefore, the angle of banking required for the curve along the highway is approximately 22.7 degrees.