Suppose the coefficient of static friction between the road and the tires on a Formula One car is 0.7 during a Grand Prix auto race. What speed will put the car on the verge of sliding as it rounds a level curve of 39.0 m radius?
To determine the speed at which the Formula One car will be on the verge of sliding as it rounds a level curve, we need to consider the maximum centripetal force that can be provided by the static friction between the tires and the road surface. The centripetal force is responsible for keeping the car moving in a curved path.
The formula to calculate the maximum static friction force is:
F_friction = μ_s * N
Where:
F_friction is the maximum static friction force,
μ_s is the coefficient of static friction, and
N is the normal force exerted on the car by the road.
The normal force can be determined using the following formula:
N = m * g
Where:
m is the mass of the car, and
g is the acceleration due to gravity.
In this case, we are given the coefficient of static friction (μ_s) as 0.7. However, to find the speed at which the car will be on the verge of sliding, we need to determine the maximum static friction force first.
The centripetal force can be calculated using the formula:
F_centripetal = (m * v^2) / r
Where:
F_centripetal is the centripetal force,
m is the mass of the car,
v is the velocity of the car, and
r is the radius of the curve.
Now, we can set the maximum static friction force equal to the centripetal force to find the velocity at which the car will be on the verge of sliding:
F_friction = F_centripetal
μ_s * N = (m * v^2) / r
Substituting the values and solving for v:
μ_s * m * g = (m * v^2) / r
v^2 = (μ_s * r * g)
v = √(μ_s * r * g)
Now, let's calculate the velocity:
Given:
μ_s = 0.7 (coefficient of static friction)
r = 39.0 m (radius of the curve)
g = 9.8 m/s^2 (acceleration due to gravity)
Plugging in the values:
v = √(0.7 * 39.0 * 9.8)
v = √(27.3 * 9.8)
v = √(267.54)
v ≈ 16.37 m/s
Therefore, the speed at which the Formula One car will be on the verge of sliding as it rounds the level curve of 39.0 m radius is approximately 16.37 m/s.
Solve this equation for V:
M V^2/R = 0.7 M g
M (the mass) cancels out, so
V = sqrt(0.7 g*R)