The sports car in Figure 6 is travelling along 30 banked road having a radius of curvature of ñ=152 m. If the coefficient of static friction between the tires and the road is µs=0.2, determine the maximum safe speed so no slipping occurs. Neglect the size of the car.
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To determine the maximum safe speed for the sports car so that no slipping occurs, we need to consider the forces acting on the car and calculate the maximum speed at which the friction force can provide enough centripetal force.
Let's start by analyzing the forces at play:
1. Gravity: The downward force due to gravity acts vertically downward and can be split into its components:
- Normal force (N): The force perpendicular to the surface of the banked road.
- Weight (mg): The force acting in the downward direction equal to the mass of the car (m) multiplied by the acceleration due to gravity (g).
2. Friction force (f): The force provided by the static friction between the tires and the road. It acts in the horizontal direction and provides the necessary centripetal force to keep the car moving in a circular path.
Since the car is not slipping, the friction force will act towards the center of the circular path and will be responsible for providing the necessary centripetal force.
The centripetal force can be calculated using the formula:
Fc = m * ac
where Fc is the centripetal force, m is the mass of the car, and ac is the centripetal acceleration.
The centripetal acceleration can be calculated using the formula:
ac = v^2 / r
where v is the velocity of the car and r is the radius of curvature of the banked road.
Now, let's calculate the maximum safe speed by equating the centripetal force to the maximum friction force:
Fc = f
m * ac = µs * N
Since the car is on a banked road, the normal force can be calculated using the following equation:
N = mg * cos(θ)
where θ is the angle of the banked road.
Additionally, we know that the friction force can be calculated using:
f = µs * N
Substituting the expressions for N and f into the equation m * ac = µs * N, we get:
m * ac = µs * mg * cos(θ)
Rearranging the equation to solve for velocity (v):
v = √(µs * r * g * cos(θ))
Now, we can substitute the given values into the equation to find the maximum safe speed.
Given: µs = 0.2, r = 152 m, and θ = 30°, and g = 9.8 m/s^2
v = √(0.2 * 152 * 9.8 * cos(30))
Using a calculator:
v ≈ √(29.568)
v ≈ 5.44 m/s
Therefore, the maximum safe speed for the sports car is approximately 5.44 m/s.