A highway curve with radius 1000ft is to be banked so that a car traveling 56.0mph will not skid sideways even in the absence of friction
At what angle should the curve be banked?
Why did the scarecrow become a civil engineer? Because he wanted to learn about bank angles and curve design! But fear not, dear traveler, for I shall assist you on this winding road. To calculate the angle at which the curve should be banked, we can employ some simple physics.
We know that for the car not to skid sideways, the net centripetal force should equal the gravitational force acting on the car. The net centripetal force can be expressed as m*v^2/r, where m is the mass of the car, v is its velocity, and r is the radius of the curve.
Now, to find the angle, we can use some trigonometry. The tangent of the angle is equal to the lateral acceleration, which is the centripetal force divided by the gravitational force or g.
So, the tangent of the angle you seek is (m*v^2)/(r*m*g). Let's plug in the values we know. Assuming your car has a mass of 1000 kg and g is 9.8 m/s^2:
angle = arctan((1000 * (56.0 m/s)^2) / (1000 ft * (0.3048 m/ft) * 1000 kg * 9.8 m/s^2))
Now, let the calculations begin, and remember to buckle up for this mathematical ride!
To determine the angle at which the curve should be banked, we can use the concept of centripetal force.
The centripetal force acting on the car as it goes around the curve is provided by the vertical component of the car's weight. This force is given by:
Fc = m * g * cos(θ)
where Fc is the centripetal force, m is the mass of the car, g is the acceleration due to gravity (approximately 32.2 ft/s^2), and θ is the angle of banking.
The centripetal force can also be expressed as:
Fc = (m * v^2) / r
where v is the velocity of the car and r is the radius of the curve.
Since we want the car to travel without skidding sideways, the force of friction is not involved. Therefore, the centripetal force and the vertical component of the car's weight are equal:
m * g * cos(θ) = (m * v^2) / r
We can cancel out the mass 'm' from both sides:
g * cos(θ) = v^2 / r
Now, we can solve for the angle θ:
cos(θ) = (v^2) / (g * r)
θ = arccos((v^2) / (g * r))
Let's substitute the given values:
v = 56.0 mph = 82.667 ft/s
r = 1000 ft
g = 32.2 ft/s^2
θ = arccos((82.667^2) / (32.2 * 1000))
Using a calculator, we find:
θ ≈ 18.97 degrees
Therefore, the curve should be banked at an angle of approximately 18.97 degrees to prevent the car from skidding sideways, assuming no friction.
To determine the angle at which the curve should be banked, we can start by considering the forces acting on the car as it moves along the curved path.
When a car travels along a banked curve, there are primarily two forces involved:
1. The gravitational force, acting vertically downward.
2. The centripetal force, which keeps the car moving in a circular path.
In the absence of friction, the centripetal force is provided solely by the horizontal component of the normal force (N) exerted by the road surface on the car.
Let's break down the gravitational force and the normal force into their respective components:
1. The gravitational force can be split into two components:
- The vertical component (Fg⊥) acts perpendicular to the road surface and is balanced by the vertical component of the normal force (N⊥).
- The horizontal component (Fg∥) acts parallel to the road surface and has no effect on providing the centripetal force.
2. The normal force can also be divided into two components:
- The vertical component (N⊥) balances the vertical component of the gravitational force (Fg⊥).
- The horizontal component (N∥) provides the centripetal force required to keep the car moving in a circular path.
Since there is no skidding sideways, we need to calculate the angle at which the horizontal component of the normal force (N∥) equals the centripetal force required for a car moving at 56.0 mph.
To find the required centripetal force, we need to convert the speed from miles per hour to feet per second since the radius is given in feet.
Converting the speed from mph to ft/s:
56.0 mph * (5280 ft/1 mile) * (1 hour/3600 s) = V ft/s (let's call it V)
The centripetal force (Fc) required to keep the car moving in a circular path is given by:
Fc = m * V^2 / r
Where:
- Fc is the centripetal force
- m is the mass of the car (not given, but we will assume it cancels out in the equation)
- V is the velocity of the car
- r is the radius of the curve
Now, equating the centripetal force to the horizontal component of the normal force, we have:
N∥ = Fc
Since we are looking for the angle at which the curve should be banked, we can use trigonometry to find the relationship between the angle (θ) and the other components.
Applying simple trigonometry, we have:
N∥ = N * sin(θ) (where N is the magnitude of the normal force)
Substituting Fc for N∥, we have:
Fc = N * sin(θ)
Rearranging for sin(θ) gives us:
sin(θ) = Fc / N
We know that sin(θ) = opposite/hypotenuse, which means:
opposite = Fc (Fc is the component along the incline)
hypotenuse = N
Therefore:
opposite/hypotenuse = Fc / N
sin(θ) = Fc / N
Now, we can calculate the angle θ by taking the inverse sine (arcsine) of the ratio Fc / N:
θ = arcsin(Fc / N)
Substituting the formula for Fc and the relationship between N and the gravitational force, we have:
θ = arcsin((V^2 / r) / (Fg⊥ + Fg∥))
Plug in the given values:
V = 56.0 mph
r = 1000 ft
Fg⊥ = m * g (where g is the acceleration due to gravity)
Please note that the value of m, the car's mass, is not provided and cancels out in the equation since it appears on both sides. Therefore, the unknown car mass will not affect the calculated angle.
So, to calculate θ, we need to know the acceleration due to gravity (g) and substitute it into the equation along with the other values.
horizontal component of g = v^2/r
g sin theta = v^2/r
here g = 32 ft/s^2
v = 56 mi/h * 5280 ft/mi * 1 h/3600 s
r = 1000 ft