## Related questions with answers

A $1050 \mathrm{~kg}$ car rounds a curve of radius $72 \mathrm{~m}$ banked at an angle of $14^{\circ}$. If the car is traveling at $85 \mathrm{~km} / \mathrm{h}$, will a friction force be required? If so, how much and in what direction?

Solution

VerifiedTo determine whether the friction is necessary or not consider the figure below. We will assume there is friction and that it points towards the center of the curve. Balancing the forces in vertical direction we have

$mg+F_{fr}\sin\theta=F_N\cos\theta.\qquad\text{(eq-1)}$

Requiring that all of the components in the direction towards the center provide for the centripetal force we have

$F_{N}\sin\theta+F_{fr}\cos\theta=\frac{mv^2}{r}.\qquad\text{(eq-2)}$

From (eq-1) we have that

$F_{N}\sin\theta\cos\theta=mg\sin\theta+F_{fr}\sin^2\theta.$

Returning this to (eq-2) premultiplied by $\cos\theta$ we get

$mg\sin\theta+F_{fr}\sin^2\theta+F_{fr}\cos^2\theta=\frac{mv^2}{r}\cos\theta.$

Using the trigonometric identity $\sin^2\theta+\cos^2\theta=1$ we find

$mg\sin\theta+F_{fr}=\frac{mv^2}{r}\cos\theta,$

yielding for friction

$F_{fr}=m\left(\frac{v^2}{r}\cos\theta-g\sin\theta\right).\qquad\text{(eq-3)}$

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