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?
forceofroad=centripetalforce
mu*mg=m*v^2/r
To determine the maximum speed at which the car can safely negotiate the curve, we need to consider the centripetal force acting on the car and compare it to the maximum friction force that can be provided by the static friction between the tires and the road.
The centripetal force is given by the formula:
Fc = (mv^2) / r
Where Fc is the centripetal force, m is the mass of the car, v is the velocity of the car, and r is the radius of curvature of the curve.
In this case, the maximum friction force is equal to the product of the coefficient of static friction (μs) and the normal force (N) acting on the car:
Ff = μs * N
The normal force N is equal to the weight of the car, which is given by:
N = m * g
Where g is the acceleration due to gravity.
To find the maximum speed, we need to equate the centripetal force and the maximum friction force:
Fc = Ff
Substituting the values, we have:
(mv^2)/r = μs * m * g
The mass 'm' cancels out from both sides of the equation:
v^2 = μs * g * r
Rearranging the equation to solve for 'v', we have:
v = sqrt(μs * g * r)
Now let's substitute the given values:
μs = 0.23 (coefficient of static friction)
g = 9.8 m/s^2 (acceleration due to gravity)
r = 75 m (radius of curvature)
Plugging in these values into the equation, we can calculate the maximum speed at which the car can safely negotiate the curve:
v = sqrt(0.23 * 9.8 * 75)
After evaluating this expression, we find that the maximum speed is approximately 24.4 m/s (or about 88 km/h).