A 14,300 N car traveling at 46.0 km/h rounds a curve of radius 2.30 x 10^2 m.
What is the minimum coefficient of static friction between the tires and the road that will allow the car to round the curve safely?
When friction is used to counteract centrifugal force, the side friction factor, f, between the tires and roadway is incorporated into the calculation for the superelevation angle. (the angle of road banking where friction is not needed)
tan@=[(Vt)^2-(f)(r)g]/[(r)g+f(Vt)^2]
(Vt)=vehicle velocity
(f)=friction coeff
(r)=radius of turn
(g)=gravity constant
If the road embankment is zero, set @=0 and solve for your friction (f)
To determine the minimum coefficient of static friction required for the car to safely round the curve, we can start by analyzing the forces acting on the car.
1. Centripetal Force: The car needs a centripetal force to maintain a circular path while rounding the curve. It is given by the equation:
F_c = (m * v^2) / r
where F_c is the centripetal force, m is the mass of the car, v is the velocity of the car, and r is the radius of the curve.
2. Frictional Force: The frictional force acting between the tires and the road provides the required centripetal force. It can be calculated using the equation:
F_friction = μ * N
where F_friction is the frictional force, μ is the coefficient of static friction, and N is the normal force acting on the car.
Since the minimum coefficient of static friction is required to safely round the curve, we can set the frictional force equal to the centripetal force. Hence:
μ * N = (m * v^2) / r
Now, we need to find the normal force N. The normal force is the force exerted by the road on the car, perpendicular to the surface of the road. In this case, it is equal to the weight of the car, which can be calculated using the formula:
N = mg
where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
With these equations, we can now solve the problem.
Given:
m = 14,300 N
v = 46.0 km/h = 46.0 m/s
r = 2.30 x 10^2 m
g = 9.8 m/s^2
1. Convert the velocity from km/h to m/s:
v = 46.0 m/s
2. Calculate the weight of the car:
N = m * g
N = 14,300 N * 9.8 m/s^2
3. Substitute the values into the equation μ * N = (m * v^2) / r:
μ * (14,300 N * 9.8 m/s^2) = (14,300 N * 46.0 m/s^2) / (2.30 x 10^2 m)
4. Rearrange the equation to solve for μ:
μ = (14,300 N * 46.0 m/s^2) / [(14,300 N * 9.8 m/s^2) * (2.30 x 10^2 m)]
5. Calculate the minimum coefficient of static friction:
μ = (14,300 N * 46.0 m/s^2) / [(14,300 N * 9.8 m/s^2) * (2.30 x 10^2 m)]
At this point, you can substitute the values into a calculator to get the final result for μ.