To determine the maximum speed at which a car can safely navigate a turn with a reversed banking angle, we need to find the speed at which the car's frictional force is equal to the centripetal force required to stay in a circular path.
To start, let's draw a free-body diagram of the car on a banked turn:
1. Identify the forces acting on the car:
- The weight of the car acting vertically downwards (mg), where m is the mass of the car and g is the acceleration due to gravity.
- The normal force (N) acting perpendicular to the surface of the road.
- The frictional force (f) acting parallel to the surface of the road, opposing the component of weight that is perpendicular to the road.
- The centripetal force (Fc) acting toward the center of the curvature.
2. Resolve the weight of the car:
The weight of the car can be resolved into two components: the component perpendicular to the road's surface (mg * cosθ) and the component parallel to the road's surface (mg * sinθ), where θ is the banking angle.
3. Determine the normal force:
The normal force is equal to the component of weight perpendicular to the road's surface. Therefore, N = mg * cosθ.
4. Calculate the frictional force:
The frictional force can be determined using the formula f = μs * N, where μs is the coefficient of static friction.
5. Determine the centripetal force:
The centripetal force is given by Fc = (mv^2) / r, where m is the mass of the car, v is the velocity of the car, and r is the radius of curvature.
6. Set up the equation:
Equating the frictional force and the centripetal force, we have μs * N = (mv^2) / r.
7. Solve for the velocity (v):
Rearrange the equation to solve for v:
v = √(μs * g * r * tanθ)
8. Plug in the known values:
Given:
μs = 0.4
m (mass) - not given
g (acceleration due to gravity) = 9.8 m/s^2
r (radius of curvature) = 13 m
θ (banking angle) = 7°
Plug in the values into the equation and calculate the maximum speed at which the car can safely navigate the turn.