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phy131studiof15:lectures:chapter6 [2015/09/14 09:08] mdawber [Banked turns] |
phy131studiof15:lectures:chapter6 [2015/09/14 09:14] mdawber [Banked turns with Friction] |
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In reality, the frictional force allows cars to round the curve within a range of safe velocities determined by the maximum frictional force that can be provided. | In reality, the frictional force allows cars to round the curve within a range of safe velocities determined by the maximum frictional force that can be provided. | ||

- | For a car that is going fast the frictional force points down the incline and stops the car from flying out, but this only works up the point where the frictional force is equal to $mu F_{N}$. So the maximum safe velocity $v_{high}$ is obtained from | + | For a car that is going fast the frictional force points down the incline and stops the car from flying out, but this only works up the point where the frictional force is equal to $\mu F_{N}$. So the maximum safe velocity $v_{high}$ is obtained from |

$F_{N}\sin\theta+\mu F_{N}\cos\theta=\frac{mv_{high}^{2}}{r}$ | $F_{N}\sin\theta+\mu F_{N}\cos\theta=\frac{mv_{high}^{2}}{r}$ | ||

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$v_{high}^{2}=rg\frac{\sin\theta+\mu\cos\theta}{\cos\theta-\mu\sin\theta}$ | $v_{high}^{2}=rg\frac{\sin\theta+\mu\cos\theta}{\cos\theta-\mu\sin\theta}$ | ||

+ | |||

+ | However it also possible to slide down the slope if a car goes too slowly. In this case friction acts up the slope and stops the car sliding down, which again only works up the point where the frictional force is equal to $\mu F_{N}$. So the minimum safe velocity $v_{low}$ is obtained from | ||

+ | |||

+ | $F_{N}\sin\theta-\mu F_{N}\cos\theta=\frac{mv_{low}^{2}}{r}$ | ||

+ | |||

+ | $F_{N}\cos\theta+\mu F_{N}\sin\theta=mg$ | ||

+ | |||

+ | $\frac{\sin\theta-\mu\cos\theta}{\cos\theta+\mu\sin\theta}=\frac{v_{low}^2}{rg}$ | ||

+ | |||

+ | $v_{low}^{2}=rg\frac{\sin\theta-\mu\cos\theta}{\cos\theta+\mu\sin\theta}$ | ||

===== 6.P.059 ===== | ===== 6.P.059 ===== | ||