Speed at which a car on a banked curve will slide.

AI Thread Summary
To determine the maximum speed at which a car can navigate a banked curve without sliding, the equations of motion and forces acting on the car are analyzed. The expressions derived include the effects of gravitational force, normal force, and static friction. An initial calculation resulted in an unreasonably high speed of 88.6 m/s, prompting a reevaluation of the equations. A correction was identified regarding the treatment of static friction, leading to a revised formula for maximum speed. The discussion emphasizes the importance of accurately applying physics principles in problem-solving.
SheldonG
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Homework Statement


If a car goes around a banked curve too fast, the car will slide out of the curve. Find an expression for the car speed v_{max} that puts the car on the verge of sliding out. What is the value for R=200, \theta = 10, \mu_s = 0.60?


Homework Equations



a = v^2/r, F = ma.


The Attempt at a Solution


Place the x-axis along the bank, and write the for equations for x and y:

\sum F_x = \frac{mv^2}{R}\cos\theta = mg\sin\theta + f_s
\sum F_y = 0 = F_n - \frac{mv^2}{R}\sin\theta - mg\cos\theta

It seems that the car should start to slide when
\frac{mv^2}{R}\cos\theta > mg\sin\theta + f_s

Using fs = \mu_s F_n

\frac{mv^2}{R}\cos\theta > mg\sin\theta + \mu_s\left(\frac{mv^2}{R}\sin\theta + mg\cos\theta\right)

Solving for v:

v > \sqrt{\frac{gR(\tan\theta + \mu_s/\tan\theta)}{1-\mu_s\tan\theta}

Unfortunately, the figure I get for v using R = 200, theta = 10, coefficient of static friction = 0.60 is much too high (88.6 m/s).

Could someone give me a hint about where I went wrong?

Thank you,
Sheldon
 
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In this term \mu_s mg\cos\theta, if one divides by cos\theta, then one does not get a tan function, but simply

\mu_s mg


So one would obtain

v > \sqrt{\frac{gR(\tan\theta + \mu_s)}{1-\mu_s\tan\theta}
 
Last edited:
Thank you, Astronuc. I am in your debt. What a stupid error :-)

Sheldon
 
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