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Car turning without skidding

  1. Jan 15, 2013 #1
    1. The problem statement, all variables and given/known data
    sNxzE.png


    2. Relevant equations
    F = ma

    3. The attempt at a solution
    First off, to keep the car from skidding to the side we want it to be constrained to move in a circle about the center of the turn because if it skids then it has deviated / veered off tangentially at some point along a circular path. For the minimum velocity, we want friction to point in the direction away from the center so that the net centripetal force is less thus requiring a minimal velocity. The equations of motion for the car are [itex]Nsin\theta - fcos\theta = M\frac{v_{min}^{2}}{R}, Ncos\theta + fsin\theta - Mg = 0[/itex] which tells us that [itex]v_{min}^{2} = gR\frac{tan\theta - \mu }{1 + \mu tan\theta }[/itex]. For the maximum case we just negate the direction of friction and this gives us, in a very similar way, [itex]v_{max}^{2} = gR\frac{tan\theta + \mu }{1 - \mu tan\theta }[/itex]. Plugging in the parameters give in the ans. clue we see that [itex]v_{min} = 0, v_{max} = \infty [/itex] so any speed in the range [itex]0 \leq v< \infty [/itex] will keep it from skidding which is all possible speeds. Is that what the text means by "all speeds are possible"? Thanks!
     
  2. jcsd
  3. Jan 15, 2013 #2

    haruspex

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