An unbanked curve has a radius of curvature of 75m. On a winter day, with snow on the road, the coefficient of static friction between the tires of the car and the road is .23. What is the maximum speed with which the car can safely negotiate the curve?
To determine the maximum speed at which the car can safely negotiate the curve, we can use the concept of centripetal force. The centripetal force is provided by the friction between the tires and the road.
The formula to calculate the centripetal force is:
F = (m * v^2) / r
Where:
F is the centripetal force
m is the mass of the car
v is the velocity of the car
r is the radius of curvature of the curve
Since we want to find the maximum speed, we need to determine the maximum value of F. In this case, the maximum value of the centripetal force occurs when it is equal to the product of the coefficient of static friction and the normal force:
F_max = μ * N
Where:
F_max is the maximum centripetal force
μ is the coefficient of static friction
N is the normal force
For a car on a level road, the normal force (N) is equal to the weight of the car (mass * gravity):
N = m * g
Where:
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Now, let's combine these equations to find the maximum speed:
F_max = μ * N
(m * v^2) / r = μ * N
Since N = m * g, we can substitute it into the equation:
(m * v^2) / r = μ * (m * g)
Simplifying the equation:
v^2 = μ * r * g
Rearranging the equation to solve for v:
v = sqrt(μ * r * g)
Now we can plug in the values given in the problem to find the maximum speed:
v = sqrt(0.23 * 75 * 9.8) ≈ 9.87 m/s
Therefore, the maximum speed with which the car can safely negotiate the curve is approximately 9.87 m/s.