a car enters a 300m radius flat curve on a rainy day when the coefficient of static friction is between its tires and the road is .600. What is the maximum speed a car can travel around a curve without sliding.
centripetal force points towards the center of the circle, and force friction points the opposite way. Set the two equal to each other
Force Normal * coefficient of static friction = mass*velocity^2/radius
(mg)Us = mv^2/r
mass isn't given in the problem because it cancels on both sides.
9.8Us = V^2/r
9.8(.600) = v^2/300
v^2 = 1764
v = 42 m/s
A car enters a 300-m radius flat curve on a rainy day when the coefficient of static friction between its tires and the road is 0.600. What is the maximum speed which the car can travel around the curve without sliding?
m/s 33.1 0
m/s 29.6 0
m/s 42
m/s 24.8
Why did the car bring an umbrella to the race track? Because it knew it was going to be a "drizzly" day! Now, let's calculate the maximum speed this "slick" car can go without sliding.
To find the maximum speed without sliding, we need to consider the centripetal force in relation to the frictional force. The equation is:
F_friction ≤ F_centripetal
μ * m * g ≤ m * v^2 / r
Where:
μ = coefficient of static friction (0.600)
m = mass of the car
g = acceleration due to gravity
v = maximum speed without sliding
r = radius of the curve (300 m)
Since mass cancels out, we can simplify the equation to:
μ * g ≤ v^2 / r
Substituting the values, we get:
0.600 * 9.8 ≤ v^2 / 300
Multiply both sides by 300 to solve for v^2:
(0.600 * 9.8) * 300 ≤ v^2
v^2 ≥ (0.600 * 9.8) * 300
v^2 ≥ 1764
So, the maximum speed the car can travel without sliding is approximately 41.96 m/s (rounded to two decimal places). Now, put on some rain boots and go "slide-free" cruising!
To calculate the maximum speed a car can travel around a curve without sliding, you need to use the formula for centripetal force:
Centripetal Force (Fc) = (mass (m) * velocity (v)^2) / radius (r)
The maximum speed a car can travel without sliding is achieved when the centripetal force equals the maximum static friction force. The maximum static friction force can be calculated using the formula:
Maximum Static Friction Force (Ffriction) = coefficient of static friction (μ) * normal force (Fnormal)
In this case, the normal force is equal to the weight (mg) of the car, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
So, the equation can be set up as follows:
Fc = Ffriction
(m * v^2) / r = μ * m * g
Canceling mass (m) on both sides:
v^2 / r = μ * g
Rearranging the equation to solve for the maximum speed (v):
v = √(μ * g * r)
Given that the radius (r) is 300m and the coefficient of static friction (μ) is 0.600, and g is approximately 9.8 m/s^2, we can calculate the maximum speed (v).
v = √(0.600 * 9.8 * 300)
v ≈ 27.04 m/s
Therefore, the maximum speed the car can travel around the curve without sliding is approximately 27.04 m/s.
To determine the maximum speed at which a car can travel around a curve without sliding, you need to use the concept of centripetal force.
First, let's understand the forces acting on the car during this maneuver. We have the gravitational force acting vertically downward, and the normal force, which is the force exerted by the road on the tires, acting vertically upward. Additionally, there is the static frictional force between the tires and the road, which acts radially inward and provides the centripetal force needed to keep the car moving in a curved path.
The maximum static frictional force can be calculated by multiplying the coefficient of static friction (μ) and the normal force (N). In this case, we don't have information about the weight of the car, so we'll use the normal force as a general variable.
The maximum static frictional force (Ff max) = μ * N.
Now, the centripetal force (Fc) acting on the car can be calculated using the formula:
Fc = (mass of the car) * (velocity squared) / (radius of the curve).
In this case, we need to isolate the centripetal force, so we can write:
Fc = Ff max.
Substituting the values, we have:
(mass of the car) * (velocity squared) / (radius of the curve) = μ * N.
Now, let's find the expression for the normal force (N). The normal force is equal to the weight of the car, which is given by:
N = (mass of the car) * (gravity).
Substituting this back into the equation, we get:
(mass of the car) * (velocity squared) / (radius of the curve) = μ * (mass of the car) * (gravity).
Now, we can cancel out the mass of the car from both sides of the equation, giving us:
(velocity squared) / (radius of the curve) = μ * (gravity).
Rearranging the equation, we get:
velocity squared = μ * (gravity) * (radius of the curve).
Finally, taking the square root of both sides gives us:
velocity = √(μ * (gravity) * (radius of the curve)).
Now, substitute the given values to find the maximum speed:
velocity = √(0.600 * (9.8 m/s²) * (300 m)).
Calculating this expression will give you the maximum speed that a car can travel around the curve without sliding.