A banked circular highway curve is designed for traffic moving at 90 km/h. The radius of the curve is 70 m. If the coefficient of static friction is 0.30 (wet pavement), at what range of minimum and maximum speeds can a car safely make the curve?
To determine the range of minimum and maximum speeds at which a car can safely make the curve, we need to consider the forces acting on the car.
The first force to consider is the gravitational force (mg) acting vertically downward. The normal force (N) acts perpendicular to the surface of the curve and counteracts the gravitational force.
The second force is the frictional force (f) acting horizontally inward, providing the centripetal force necessary to keep the car moving in a circular path. The maximum frictional force can be calculated using the formula f = µN, where µ is the coefficient of static friction.
Since the car is moving along the curve, we can equate the maximum frictional force to the centripetal force needed to keep the car moving in a circular path.
The centripetal force (Fc) is given by the equation Fc = m(v^2/r), where m is the mass of the car, v is the velocity, and r is the radius of the curve.
Now, let's calculate the maximum frictional force:
f = µN
= µmg
To calculate the centripetal force:
Fc = m(v^2/r)
Since we are interested in finding the minimum and maximum speeds, we can set up an inequality using the maximum frictional force and centripetal force:
µmg ≥ m(v^2/r)
Canceling out the mass (m) on both sides of the inequality, we get:
µg ≥ v^2/r
Rearranging the equation to solve for v, the velocity:
v^2 ≤ µgr
Taking the square root of both sides, we get:
v ≤ √(µgr)
Therefore, the car can safely make the curve if its velocity (v) is less than or equal to the square root of µgr.
To find the minimum and maximum speeds, substitute the given values into the equation:
v ≤ √(0.30 * 9.8 * 70)
Calculating this expression:
v ≤ √(20.58)
So, the car can safely make the curve if its speed is less than or equal to approximately 4.54 m/s.
Now, to convert this speed to km/h:
4.54 m/s * 3.6 km/h = 16.34 km/h
Therefore, the range of minimum and maximum speeds at which the car can safely make the curve is approximately 0 km/h to 16.34 km/h.