If the posted speed limit for a particular curve of radius 36 m is 15.7 m/s at what angle should the road be banked so that cars will stay on a circular path even if there were no friction between the road and the tires?
To find the angle at which the road should be banked, we need to determine the force components acting on the car while it is moving around the curve.
First, let's determine the speed of the car when it is moving on the curved road. We know that the posted speed limit is 15.7 m/s, which is also the speed of the car.
Using the formula for centripetal acceleration, we have:
a = v^2 / r
Where:
a = centripetal acceleration
v = velocity of the car
r = radius of the curve
Plugging in the values, we get:
a = (15.7 m/s)^2 / 36 m
a ≈ 6.825 m/s^2
Now, we need to find the net force acting on the car. In this case, the net force is the resultant force that should be provided by the banking angle to make the car follow the curved path.
The net force can be split into two components: the vertical component (perpendicular to the road surface) and the horizontal component (parallel to the road surface).
The vertical component of the net force is the force due to gravity acting on the car. The equation for this force is:
F_gravity = m * g
Where:
F_gravity = force due to gravity
m = mass of the car
g = acceleration due to gravity (approximately 9.8 m/s^2)
The horizontal component of the net force provides the centripetal force required to keep the car on the curved path. The equation for this force is:
F_centripetal = m * a
Where:
F_centripetal = centripetal force (horizontal component of the net force)
Since we want the car to stay on the circular path even if there is no friction, the horizontal component of the net force should be equal to the centripetal force.
Now, let's consider the forces acting on the car when it is moving on the curved road. The normal force (N) splits into two components: the vertical component (N_vert) and the horizontal component (N_hor).
The vertical component of the normal force (N_vert) balances the force due to gravity:
N_vert = m * g
The horizontal component of the normal force (N_hor) provides the horizontal component of the net force required for the car to follow a circular path:
N_hor = F_centripetal = m * a
The angle of inclination (θ) of the road is defined as the angle made between the horizontal surface and the direction of the net force exerted by the road on the car.
To calculate the angle (θ), we can equate the two components of the normal force:
N_vert / N_hor = tan(θ)
Therefore:
m * g / (m * a) = tan(θ)
Finally, we can solve for the angle (θ):
θ = atan(g / a)
Plugging in the values:
θ = atan(9.8 m/s^2 / 6.825 m/s^2)
θ ≈ atan(1.435)
θ ≈ 55.5°
Therefore, the road should be banked at an angle of approximately 55.5° so that cars will stay on a circular path even if there were no friction between the road and the tires.