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Chapter 5: Problem 19

A curve in a stretch of highway has radius \(R .\) The road is unbanked. The coefficient of static friction between the tires and road is \(\mu_{s} .\) (a) What is the fastest speed that a car can safely travel around the curve? (b) Explain what happens when a car enters the curve at a speed greater than the maximum safe speed. Illustrate with an FBD.

Short Answer

Expert verified
Answer: The maximum safe speed a car can travel around a curve is given by the equation v = sqrt(碌s * g * R). If the car enters the curve at a speed greater than the maximum safe speed, the frictional force between the tires and the road will not be enough to provide the required centripetal force, causing the car to skid or slide off the curve.

Step by step solution

01

Analyzing Maximum Safe Speed

Determine the maximum speed at which a car can travel around the curve while maintaining the static friction between the tires and the road. Centripetal force (Fc) is responsible for keeping the car moving in a circular path: Fc = (mv^2) / R where m is the mass of the car, v is its speed, and R is the radius of the curve.
02

Determine Frictional Force

Calculate the maximum frictional force (Ff) that can act between the car's tires and the road: Ff = 碌s * Fn where 碌s is the coefficient of static friction and Fn is the normal force acting on the car, which is equal to the car's weight (mg) in this unbanked situation.
03

Compare Frictional Force and Centripetal Force

In order for the car to maintain its circular path, the centripetal force must be equal to the frictional force: (mv^2) / R = 碌s * mg
04

Solve for Maximum Safe Speed (v)

Now we can solve for the maximum safe speed (v): v^2 = 碌s * mgR v = sqrt(碌s * g * R)
05

Answer (a)

The fastest speed that a car can safely travel around the curve is given by: v = sqrt(碌s * g * R)
06

Answer (b)

If a car enters the curve at a speed greater than the maximum safe speed, the frictional force between the tires and the road will not be enough to provide the required centripetal force. As a result, the car will skid or slide off the curve. An FBD of this situation would show the centripetal force no longer equal to the frictional force, with the car no longer maintaining a circular path and instead moving off of the curve.

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