"An unbaked 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 care can safely negotiate the cure?"
r=75 and static firction=.23 is all that is give, please show steps to solution.
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 given by the equation Fc = mv²/r, where Fc is the centripetal force, m is the mass of the car, v is the speed of the car, and r is the radius of curvature.
In this case, we need to find the maximum speed that the car can safely negotiate the curve. To do that, we first need to calculate the maximum value of the static friction force that can be exerted by the tires. The formula for static friction is Fs ≤ μsN, where Fs is the static friction force, μs is the coefficient of static friction, and N is the normal force.
Since the car is on a curve, the normal force will be equal to the weight of the car, N = mg, where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s²).
Now we can substitute the value of Fs into the centripetal force equation:
Fs = mv²/r.
So, we have μsN = mv²/r.
Substituting the value of N:
μs(mg) = mv²/r.
Canceling out the mass on both sides gives:
μsg = v²/r.
To find the maximum speed, we rearrange the equation to solve for v:
v = √(μsg * r).
Plugging in the given values:
v = √(0.23 * 9.8 * 75).
Calculating this expression results in:
v ≈ 14.41 m/s.
Therefore, the car can safely negotiate the curve on a winter day with a maximum speed of approximately 14.41 m/s or about 52 km/h.