A 1400-kg car rounds a curve of 57-m banked at an angle of 14 degrees.
If the car is traveling at 98-km/h (27-m/s), will a friction force be required? If so in what direction?
How much friction force will be required?
Let the x-axis point toward the center of curvature and the y-axis point upward. Use Newton’s second law.
ΣFy = N• cosθ − mg − f •sinθ = 0 ...(1)
ΣFx = N• sinθ + f •cosθ = m•v²/R,....(2)
Solving the 1st equation for N, we obtain
N =( f •sin θ + m•g)/cos θ,
Substitute N to the 2nd equation and obtain friction force
f =m[(v²•cosθ/R) - g•sinθ].
f=[fy.fx].N
this sucks. no help at all
To determine if a friction force is required in this scenario, we need to consider the forces acting on the car. In this case, the forces involved are the gravitational force (mg), the normal force (N), and the friction force (f).
First, let's calculate the normal force acting on the car. The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is directed perpendicular to the banked curve and opposes the gravitational force.
The formula to calculate the magnitude of the normal force is given by:
N = mg cos(θ)
Where:
m = mass of the car (1400 kg)
g = acceleration due to gravity (9.8 m/s^2)
θ = angle of the bank (14 degrees)
Substituting the given values into the equation:
N = (1400 kg) x (9.8 m/s^2) x cos(14°)
Next, let's calculate the friction force required. The friction force opposes the car's tendency to slide down the slope. To determine if a friction force is required, we need to compare the gravitational force component parallel to the slope (mg sin(θ)) with the centripetal force (mv^2/r). If the centripetal force is greater than the gravitational force component, a friction force is required.
The formula to calculate the magnitude of the centripetal force is given by:
Fc = mv^2/r
Where:
m = mass of the car (1400 kg)
v = velocity of the car (27 m/s)
r = radius of the curve (57 m)
Substituting the given values into the equation:
Fc = (1400 kg) x (27 m/s)^2 / 57 m
Now we can compare the gravitational force component (mg sin(θ)) with the centripetal force (Fc). If Fc is greater, a friction force will be required, and its direction will be opposite to the car's motion. If Fc is smaller or equal, no friction force is needed.
Finally, if a friction force is required, we can calculate its magnitude by subtracting the gravitational force component from the centripetal force:
f = Fc - mg sin(θ)
Substituting the known values into the equation:
f = [(1400 kg x (27 m/s)^2) / 57 m] - (1400 kg x 9.8 m/s^2 x sin(14°))
By performing these calculations, we can determine if a friction force is required and calculate its magnitude.