Question Dealing with Newtons Laws

[glockstock]

Homework Statement

A curve of radius 16 m is banked so that a 940 kg car traveling at 44.4 km/h can round it even if the road is so icy that the coefficient of static friction is approximately zero. You are commissioned to tell the local police the range of speeds at which a car can travel around this curve without skidding. Neglect the effects of air drag and rolling friction. If the coefficient of static friction between the road and the tires is 0.300, what is the range of speeds you tell them?

Homework Equations

Fnet = angular vel. * r * m

The Attempt at a Solution

I got vmax right = 59.69 m/s

vmin = ?

 

[HallsofIvy]

If the car were going too fast, it would slide up the banked turn. If the car were going too slow, its weight would make it slide down the slope of the banked turn.

 

[tomcue]

Based on the given information, the range of speeds at which a car can travel around the curve without skidding can be calculated using Newton's laws of motion. We can use the equation for centripetal force, Fc = mv^2/r, where m is the mass of the car, v is the velocity, and r is the radius of the curve. This force must be balanced by the force of static friction acting on the car, which is given by Fs = μsN, where μs is the coefficient of static friction and N is the normal force.

At the maximum velocity (vmax), the force of static friction must be equal to the maximum possible value, which is μsN = μsmg. Therefore, we can set the equations for centripetal force and static friction equal to each other and solve for vmax:

Fc = Fs
mvmax^2/r = μsmg
vmax = √(μsrg/m)

Plugging in the given values, we get vmax = √(0.300*16*9.8/940) = 8.86 m/s. Converting this to km/h, we get vmax = 31.9 km/h.

At the minimum velocity (vmin), the force of static friction must be equal to zero. Therefore, we can set the equation for centripetal force equal to zero and solve for vmin:

Fc = 0
mvmin^2/r = 0
vmin = 0 m/s

Therefore, the range of speeds at which the car can travel around the curve without skidding is 0 m/s to 8.86 m/s, or 0 km/h to 31.9 km/h. It is important to note that these values assume there is no air drag or rolling friction present. In real-world situations, these factors may affect the speed range and should be taken into consideration.