Max and min speed around banked curve

AI Thread Summary
To determine the maximum and minimum speeds for a car navigating a banked curve without skidding, one must analyze the forces acting on the vehicle, including gravitational force, normal force, and friction. The equations of motion involve calculating the normal force (N) as mg cos(Theta) and the friction force (f) as muN. The conditions for slipping up or down the curve require examining the balance of forces, where the centripetal acceleration (ac) is given by v^2/R. By setting up equations for both scenarios—slipping up and slipping down—the relationship between speed, radius, and banking angle can be established. Properly relating these forces and conditions will yield the required speed limits for safe navigation of the turn.
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Homework Statement



A car enters a turn whose radius is R. The road is banked at angle Theta, and the friction coefficient is mu. Find the max and min speeds for the car to stay on the road without skidding sideways.

Homework Equations



W = mg
N = -W
Friction force = f = muN
Centripetal acceleration = ac = v^2/R

The Attempt at a Solution


Here's the force diagram I drew:
http://img504.imageshack.us/img504/9576/0925081412wy0.th.jpg http://g.imageshack.us/thpix.php

I have the following equations set up:
N = mgcosTheta f = mumgcosTheta W = mg
-W + N + f = 0
NcosTheta - fcosTheta = Fc = ac

So far, I think I've set up the problem correctly, but now I don't know where to go from here.
 
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diablo2121 said:

Homework Statement



A car enters a turn whose radius is R. The road is banked at angle Theta, and the friction coefficient is mu. Find the max and min speeds for the car to stay on the road without skidding sideways.

Homework Equations



W = mg
N = -W
Friction force = f = muN
Centripetal acceleration = ac = v^2/R

The Attempt at a Solution


Here's the force diagram I drew:

I have the following equations set up:
N = mgcosTheta f = mumgcosTheta W = mg
-W + N + f = 0
NcosTheta - fcosTheta = Fc = ac

So far, I think I've set up the problem correctly, but now I don't know where to go from here.

Examine the cases separately. For instance at the velocity that it would slip up the curve, what must the V be greater than?

Likewise for the case where it would slip down the inclined curve what does the force drawing tell you about how to treat the terms?
 
If the car were to slip up the curve, then the friction force f, pointing inward, is overcome. Likewise, the car slipping inwards would imply that W is overcoming f, which should be pointing outwards. How do I relate these concepts into my equations?
 
diablo2121 said:
If the car were to slip up the curve, then the friction force f, pointing inward, is overcome. Likewise, the car slipping inwards would imply that W is overcoming f, which should be pointing outwards. How do I relate these concepts into my equations?

Just write it down. What is the normal force? What force opposes it for it to slip in one direction or the other. What do you do with the weight component that is the sinθ term? What must the velocity be greater or less than for each case?
 
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