Chapter 5: Problem 48
A flat (unbanked) curve on a highway has a radius of 170.0 m. A car rounds the curve at a speed of 25.0 m/s. (a) What is the minimum coefficient of static friction that will prevent sliding? (b) Suppose that the highway is icy and the coefficient of static friction between the tires and pavement is only one- third of what you found in part (a). What should be the maximum speed of the car so that it can round the curve safely?
Short Answer
Step by step solution
Understanding the Forces
Calculate Required Centripetal Force
Express Static Friction Force
Set Up the Equation for Static Friction
Calculate Minimum Coefficient of Static Friction
Adjust for Reduced Coefficient
Calculate Maximum Safe Speed on Ice
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Centripetal Force
To compute the centripetal force (\[ f_c \]), we use the formula \[ f_c = \frac{mv^2}{r} \]. In this equation, \( m \) represents the mass of the car, \( v \) is its speed, and \( r \) is the radius of the curve. Although the mass doesn't directly affect the determination of the required coefficient of friction, it's crucial to note that this force essentially compels the car to seek the center of the circle as it moves around the curve.
In the exercise, the centripetal force is counteracted by the static frictional force coming from the contact between the car's tires and the road. If the frictional force weren't present or sufficient, the car would fail to follow the curve and slide away from the circular path.
Coefficient of Friction
The static frictional force (\( f_s \)) is expressed as \[ f_s = \mu_s F_n \], where \( F_n = mg \) is the normal force, or the perpendicular force that a surface exerts on an object. For the car not to slide off the curve, this static frictional force must be at least equal to the centripetal force. Thus, we have \( \mu_s mg = \frac{mv^2}{r} \), leading to \( \mu_s = \frac{v^2}{rg} \).
In scenarios where surfaces may be less conducive to friction, such as an icy road, the coefficient drops. Consequently, the speed at which the car can handle the turn safely decreases. This concept is crucial, not just for understanding physics problems, but also for real-world driving safety.
Circular Motion
For real-world applications, the role of static friction in providing enough grip to allow vehicles to complete such curves cannot be overstated. The static friction must match or exceed the force needed to change the vehicle's straightforward velocity into a circular path.
- Vehicles must adjust speed, as determined by \( v = \sqrt{\mu r g} \), to safely maneuver around centers with differing coefficients of friction.
- The concept demonstrates the delicate balance necessary for achieving safe transit during turns.