If a curve with a radius of 91.5 m is perfectly banked for a car traveling 77.4 km/h, what must be the coefficient of static friction for a car not to skid when traveling at 91.6 km/h?
To solve this problem, we need to consider the forces acting on the car as it goes around the banked curve.
The first force to consider is the gravitational force (weight) acting vertically downward. The weight can be decomposed into two components: the normal force (N) acting perpendicular to the surface of the curve, and the force of gravity acting vertically downward.
The second force to consider is the frictional force (F) acting horizontally inwards on the car. This force prevents the car from sliding or skidding off the curve.
In order for the car not to skid, the frictional force must provide the necessary centripetal force to keep the car moving in a circular path. The centripetal force is given by the equation:
F_c = (m * v^2) / r
where F_c represents 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.
Now, let's calculate the centripetal force for the first scenario, where the car is traveling at 77.4 km/h:
Convert the velocity from km/h to m/s:
v = 77.4 km/h * (1000 m/1 km) * (1 h/3600 s) = 21.5 m/s
Substituting the values into the equation for centripetal force:
F_c = (m * v^2) / r
Next, let's calculate the centripetal force for the second scenario, where the car is traveling at 91.6 km/h:
Convert the velocity from km/h to m/s:
v = 91.6 km/h * (1000 m/1 km) * (1 h/3600 s) = 25.4 m/s
Now, to find the coefficient of static friction (μ) for the car not to skid, we need to compare the centripetal force in the first scenario with the maximum frictional force that can be provided:
F_max = μ*N
Since the car is not rotating or sliding vertically, the normal force is equal to the weight of the car:
N = mg
Substituting the weight into the maximum frictional force equation:
F_max = μ * mg
To balance the forces in the horizontal direction, the maximum frictional force must equal the centripetal force:
F_max = F_c
μ * mg = (m * v^2) / r
Finally, we can solve for the coefficient of static friction (μ):
μ = (v^2) / (g * r)
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now, plug in the values to find the coefficient of static friction required for the car not to skid:
μ = (25.4 m/s)^2 / (9.8 m/s^2 * 91.5 m)