Two curves on a highway have the same radii. However, one is unbanked and the other is banked at an angle θ. A car can safely travel along the unbanked curve at a maximum speed v0 under conditions when the coefficient of static friction between the tires and the road is μs = 0.423. The banked curve is frictionless, and the car can negotiate it at the same maximum speed v0. Find the angle θ of the banked curve.
Well, isn't this quite the engineering problem! Let's see if we can find the angle θ.
First, let's take a look at the unbanked curve. Since there is no friction involved, the only force acting on the car is the centripetal force. In order for the car to stay on the curve, the centripetal force must be provided by the normal force, which is equal to the car's weight.
So, on the unbanked curve, we have:
m * g = m * v0^2 / R
Where m is the mass of the car, g is the acceleration due to gravity, v0 is the maximum speed, and R is the radius of the curve.
Now, let's move on to the banked curve. Here, we have two forces at play: the weight of the car pulling it downward, and the component of the normal force perpendicular to the surface of the road providing the centripetal force.
The normal force can be broken down into two components: the vertical component (n * cos(θ)) and the horizontal component (n * sin(θ)).
The horizontal component of the normal force provides the centripetal force, so we have:
m * v0^2 / R = n * sin(θ)
The vertical component of the normal force balances the weight of the car, so we have:
m * g = n * cos(θ)
Now, let's use the fact that the normal force is equal to the weight of the car multiplied by the coefficient of static friction (μs):
n = m * g * μs
Plugging this into the equations above, we get:
m * v0^2 / R = m * g * μs * sin(θ)
m * g = m * g * μs * cos(θ)
Canceling out the mass from both sides, we have:
v0^2 / R = g * μs * sin(θ)
g = g * μs * cos(θ)
Dividing the second equation by the first equation, we get:
(v0^2 / R) / g = (g * μs * cos(θ)) / (g * μs * sin(θ))
Simplifying, we have:
v0^2 / R = cos(θ) / sin(θ)
Now, we can use some trigonometric identities to simplify further. Remembering that cos(θ) / sin(θ) is equal to cot(θ), we have:
v0^2 / R = cot(θ)
Finally, solving for θ:
θ = arctan(v0^2 / R)
So, to find the angle θ of the banked curve, we need to take the arctan of (v0^2 / R).
To find the angle θ of the banked curve, we can use the concept of forces acting on the car while negotiating the banked curve.
Let's consider the forces acting on the car while it traverses the banked curve:
1. Gravitational force (mg): This force acts vertically downwards and can be resolved into two components: one parallel to the inclined surface (mg*sinθ) and the other perpendicular to the inclined surface (mg*cosθ).
2. Centripetal force (Fc): This force acts towards the center of the circular path and allows the car to turn. Its magnitude can be given by Fc = (mv^2) / r (where m is the mass of the car, v is the velocity, and r is the radius of the curve).
Since the car can negotiate the banked curve at the same maximum speed (v0) as the unbanked curve, we can equate the magnitudes of the centripetal forces on both curves:
Fc (unbanked) = Fc (banked)
(mv0^2) / r (unbanked) = (mv0^2) / r (banked)
Now, let's consider the friction force acting on the car while it negotiates the unbanked curve:
3. Friction force (f): This force acts horizontally towards the center of the curve and provides the necessary centripetal force on the unbanked curve. Its magnitude can be given by f = μs * N, where N is the normal force acting on the car.
To find the normal force, we need to consider the vertical forces on the car:
1. mg*cosθ (downwards)
2. Normal force (N) (upwards)
For the car to remain in equilibrium in the vertical direction, the downward and upward forces must balance:
mg * cosθ = N
Since the friction force provides the necessary centripetal force on the unbanked curve, the friction force must be equal to the centripetal force:
f = Fc (unbanked)
μs * N = (mv0^2) / r (unbanked)
Substituting the value of N from the equilibrium equation:
μs * (mg * cosθ) = (mv0^2) / r (unbanked)
Now, let's consider the frictionless banked curve.
Since there is no friction force on the banked curve, the only vertical force acting on the car is the gravitational force component parallel to the inclined surface: mg * sinθ.
Again, for the car to remain in equilibrium in the vertical direction, this downward force must be balanced by the normal force:
mg * sinθ = N
Substituting the value of N in terms of mg * sinθ:
μs * (mg * cosθ) = (mv0^2) / r (unbanked)
mg * sinθ = (mv0^2) / r (banked)
Now, we can equate the two equations:
μs * (mg * cosθ) = mg * sinθ
μs * cosθ = sinθ
θ = arctan(μs)
Substituting the given value of μs = 0.423:
θ ≈ arctan(0.423)
Therefore, the angle θ of the banked curve is approximately equal to arctan(0.423).
To find the angle θ of the banked curve, we can start by analyzing the forces acting on the car as it negotiates the banked curve.
On the banked curve, the only force acting on the car is its weight, which can be resolved into two components: one perpendicular to the surface of the curve (N⊥) and one parallel to the surface (N∥). The perpendicular component N⊥ provides the necessary centripetal force for the car to travel in a circular path.
The maximum speed the car can travel on the unbanked curve is determined by the coefficient of static friction between the tires and the road, which is given as μs = 0.423. This friction force (fs) provides the necessary centripetal force.
On the unbanked curve, the force fs can be calculated using the equation:
fs = μs * N⊥
Since N⊥ is equal to the weight of the car (mg), where m is the mass of the car and g is the acceleration due to gravity, we can substitute it in the equation:
fs = μs * mg
On the banked curve, there is no friction force, and the only force providing the centripetal force is the perpendicular component of the weight N⊥.
Using trigonometry, we can relate N⊥ to the weight of the car (mg) and the angle θ. The equation is:
N⊥ = mg * cos(θ)
Since the car can safely negotiate both curves at the same maximum speed v0, the centripetal forces on both curves should be equal. Therefore, we can equate the expressions for fs and N⊥:
fs = μs * mg (equation 1)
N⊥ = mg * cos(θ) (equation 2)
By equating equation 1 and equation 2, we can solve for θ:
μs * mg = mg * cos(θ)
Dividing both sides by mg, we get:
μs = cos(θ)
Finally, we can solve for θ by taking the inverse cosine (cos^-1) of both sides of the equation:
θ = cos^-1(μs)
Substituting the given value of μs = 0.423, we can find the angle θ of the banked curve.