1300 kg car rounds a banked curve of radius= 50m. If car traveling at 75 km/h and the coefficient of static friction between the tire and road is 0.45 ...
1.) what is the minimum angle of the bank needed to keep the car in the turn?
2.) If the car changes tires, which increases the Coef. of Friction to 0.65.
3.) What will be the max. Velocity for the car to be able to turn (assuming the same banked angle)?
This one is almost impossible to do without the FBD in front of you. In this problem the mg force is down the normal is purpendicular to the track and friction is parallel to the track.
Equation in y:
Fn cosθ + Ff sinθ = mg
Equation in x (causing circ motion):
Fn sinθ - Ff cosθ = mv^2/r
Third eq you need mu Fn = Ff
Now you have three eqs and three unknowns.The math is less than delightful.
Sorry, that should be
y direction:
Fn cosθ = mg + Ffsinθ
x direction:
Fn sinθ + Ff cosθ = mv^2/r
And:
mu Fn = Ff
We're trying to not fly off the track.
I get a value of about 17.1o which seems reasonable.
How did you get 17.1
To find the answers to these questions, we will use the concepts of centripetal force, gravitational force, and frictional force.
First, let's find the minimum angle of the bank needed to keep the car in the turn.
1.) To find the minimum angle of the bank, we need to equate the gravitational force and the net force acting towards the center of the curve.
The gravitational force acting on the car can be calculated using the formula:
Gravitational force = mass * gravitational acceleration
Gravitational acceleration is approximately equal to 9.8 m/s^2.
Given that the mass of the car is 1300 kg, the gravitational force can be calculated as follows:
Gravitational force = 1300 kg * 9.8 m/s^2
Next, we need to calculate the net force acting towards the center of the curve using the formula:
Net force = Frictional force + Component of gravitational force perpendicular to the inclined surface
The component of gravitational force perpendicular to the inclined surface can be calculated using the formula:
Component of gravitational force perpendicular to inclined surface = Gravitational force * cos(angle)
Now, we can equate the net force to the centripetal force.
Centripetal force = (mass * velocity^2) / radius
Let's equate the two forces and solve for the angle:
Net force = Centripetal force
(Frictional force + Component of gravitational force perpendicular to the inclined surface) = (mass * velocity^2) / radius
Substituting the given coefficient of static friction between the tire and road:
(Frictional force + Component of gravitational force perpendicular to the inclined surface) = (mass * velocity^2) / radius
Frictional force = coefficient of static friction * (Component of gravitational force perpendicular to the inclined surface)
Then, substitute the formula for the gravitational force component:
Frictional force = coefficient of static friction * (Gravitational force * cos(angle))
Finally, solve for the angle:
coefficient of static friction * (Gravitational force * cos(angle)) + (Gravitational force * sin(angle)) = (mass * velocity^2) / radius
Now, you can solve this equation using numerical methods such as iteration or graphing the equation to find the angle that satisfies the equation.