In the study of centripetal forces, a specific scenario known as a flat curve is essential to understand. This occurs when an object, such as a car, travels in a circular path, requiring a force to maintain that motion. A common real-life example is driving around a roundabout. To analyze the maximum speed at which a car can navigate a flat curve without slipping, we can apply the principles of static friction and centripetal acceleration.
Consider a car with a mass of 800 kilograms rounding a curve with a radius of 50 meters. The coefficient of static friction between the tires and the road is given as 0.5. The goal is to determine the maximum speed, denoted as \( v_{\text{max}} \), that the car can achieve while maintaining traction.
As the car moves in a circle, it experiences centripetal acceleration, which can be expressed with the formula:
\[ a_c = \frac{v^2}{r} \]
where \( a_c \) is the centripetal acceleration, \( v \) is the velocity, and \( r \) is the radius of the curve. To find \( v_{\text{max}} \), we need to analyze the forces acting on the car using a free body diagram.
In this scenario, the forces include the weight of the car (\( mg \)) acting downward and the normal force from the road acting upward. The force responsible for providing the necessary centripetal acceleration is the static friction force, which can be expressed as:
\[ f_s^{\text{max}} = \mu_s \cdot N \]
where \( \mu_s \) is the coefficient of static friction and \( N \) is the normal force. Since the only vertical forces are the weight and the normal force, they must balance each other, leading to:
\[ N = mg \]
Substituting this into the static friction equation gives:
\[ f_s^{\text{max}} = \mu_s \cdot mg \]
Setting the maximum static friction equal to the required centripetal force, we have:
\[ \mu_s \cdot mg = \frac{mv^2}{r} \]
Here, the mass \( m \) cancels out, simplifying our equation to:
\[ \mu_s \cdot g = \frac{v^2}{r} \]
Rearranging this yields the expression for the maximum speed:
\[ v^2 = g \cdot r \cdot \mu_s \]
Substituting the known values—where \( g \) (acceleration due to gravity) is approximately 9.8 m/s², \( r = 50 \) m, and \( \mu_s = 0.5 \)—we can calculate:
\[ v^2 = 9.8 \cdot 50 \cdot 0.5 \]
\[ v^2 = 245 \]
Taking the square root gives:
\[ v_{\text{max}} = \sqrt{245} \approx 15.6 \text{ m/s} \]
This result indicates that if the car exceeds a speed of 15.6 m/s while navigating the curve, the tires will begin to slip, transitioning from static to kinetic friction, which could lead to loss of control. Understanding this relationship between speed, friction, and centripetal force is crucial for safe driving practices on curved paths.
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