Circular Motion and coefficient of static friction

[kbrowne29]

I've been having trouble with the following problem:

A curve of radius 60 m is banked for a design speed of 100 km/hr. If the coefficient of static friction is .30, at what range of speeds can a car safely make the curve?

Here's what (I think) I know:
We know that there are two forces that are acting towards the center of the circle. One is friction, equal to mu * normal force. The noraml force is equal to mg, and so friction is equal to mu*mg. The other force acting towards the center of the circle is the centripetal force, which is equal to mv^2/r.
So...the maximum speed a car could have without skidding out would be given solving for v in the equation mu*mg=mv^2/r. However, I don't know what to do after this, if this is even right (which it probably isn't). I would appreciate any help with this problem. Thanks.

 

[NateTG]

First off, you forgot that the normal force has a component to the inside of the curve since it is banked.

There are two ways that a car can fail to negotiate the turn:
1. The car slides out of the curve from going to fast. In this case, friction is acting with the normal force to provide centripetal acceleration.

2. The car slips into the middle because the curve is too steep. In this case, friction is acting against the normal force which is providing more than the centripetal acceleration.

These two scenarios should give you minimum and maximum values for v.

 

[M98Ranger]

Firstly, you're on the right track with identifying the two forces acting towards the center of the circle: friction and centripetal force. The maximum speed that a car can have without skidding out is given by the equation you mentioned, where the friction force (mu*mg) is equal to the centripetal force (mv^2/r). This is because at this speed, the two forces are in equilibrium and the car will be able to safely make the curve without slipping or sliding.

Now, to find the range of speeds at which the car can safely make the curve, we need to consider the different scenarios where the friction force is greater or less than the centripetal force.

If the speed of the car is lower than the maximum speed (calculated using the equation above), then the friction force will be greater than the centripetal force. In this case, the car will be able to safely make the curve without any issues.

On the other hand, if the speed of the car is higher than the maximum speed, then the centripetal force will be greater than the friction force. This means that the car will start to slip or slide on the curve, as it is not able to generate enough friction to counteract the centripetal force. This is where the coefficient of static friction comes into play.

The coefficient of static friction represents the maximum amount of friction that can be generated between two surfaces before they start to slip or slide against each other. In this case, the coefficient of static friction is given as 0.30. This means that the maximum speed at which the car can safely make the curve is when the friction force (mu*mg) is equal to the maximum possible value of mu*N, which is equal to 0.30*mg.

So, to find the range of speeds at which the car can safely make the curve, we need to solve the equation mu*mg=mv^2/r for v, where mu is 0.30 and r is 60 m. This will give us the maximum speed that the car can have without slipping or sliding on the curve. Any speed lower than this value will be safe for the car to make the curve.

In summary, the range of speeds at which the car can safely make the curve is from 0 km/hr to the maximum speed calculated using the equation mu*mg=mv^2/r, where mu is 0.30 and r is