# f a car takes a banked curve at less than the ideal speed, friction is

f a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a real problem on icy mountain roads). (a) Calculate the ideal speed to take a 80 m radius curve banked at $15.0\cdot$. (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 25.0 km/h?

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Charles Benedict
Given data:

$\theta ={15}^{\circ }$
a) the ideal speed of car ${v}_{1}$
The velocity ${v}_{1}$ is given by
$\mathrm{tan}\theta =\frac{{v}_{1}^{2}}{r\cdot g}$
${v}_{1}=\sqrt{r\cdot g\cdot \mathrm{tan}\theta }$
${v}_{1}=\sqrt{80×9.8×\mathrm{tan}\left(15\right)}$

b) where the coefficient of friction $\mu$
We know force of friction of given by
$f={\mu }_{k}\cdot N={\mu }_{k}\cdot m\cdot g$
the centripetal force acting on the car for speed ${v}_{1}$

the centripetal force acting on the car for speed

the frictional force acting due to the centripetal force
$f=|{F}_{1}-{F}_{2}|$
on substituting the respective values

${\mu }_{k}=\frac{2.02}{9.8}=0.206$
${\mu }_{k}=0.206$