You are designing a curve in a highway. If the turn is too sharp (the radius of curvature is too small), then cars may lose traction. Find the minimum radius of curvature of a road so that a 1000kg car doesn't skid. The speed limit is 15m/s and the static friction between the tires and the road is 2000N.
frictionforce=centripetalforce
2000=1000*15^2/r
r=225/2=....
To find the minimum radius of curvature of the road, we need to consider the forces acting on the car as it turns.
In this scenario, the maximum static friction force provides the centripetal force required to keep the car moving in a circular path without skidding. The centripetal force is given by the equation:
Fc = m * v^2 / r
Where:
- Fc is the centripetal force
- m is the mass of the car (1000 kg in this case)
- v is the speed of the car (15 m/s in this case)
- r is the radius of curvature of the road (what we want to find)
The maximum static friction force is given by:
Ffriction = μs * N
Where:
- Ffriction is the maximum static friction force
- μs is the static friction coefficient between the tires and the road (2000 N in this case)
- N is the normal force acting on the car (equal to the weight of the car, which is m * g, where g is the acceleration due to gravity, approximately 9.8 m/s^2)
Now, since the maximum static friction force provides the centripetal force, we can equate the two equations:
μs * N = m * v^2 / r
Substituting the values we know:
2000 N = 1000 kg * (15 m/s)^2 / r
Simplifying the equation, we can solve for the minimum radius of curvature:
r = 1000 kg * (15 m/s)^2 / 2000 N
r = 112.5 meters
Therefore, the minimum radius of curvature of the road should be 112.5 meters to prevent the 1000 kg car from skidding at a speed of 15 m/s, given the maximum static friction coefficient between the tires and the road is 2000 N.