A circular racetrack of radius $300\; m$ is banked at an angle of $15^o$. If the coefficient of friction between the wheels of a race-car and the road is $0.2$, what is the
$(a)$ optimum speed of the racecar to avoid wear and tear on its tyres, and
$(b)$ maximum permissible speed to avoid slipping ?
A circular racetrack of radius $300\; m$ is banked at an angle of $15^o$. If the coefficient of friction between the wheels of a race-car and the road is $0.2$, what is the
$(a)$ optimum speed of the racecar to avoid wear and tear on its tyres, and
$(b)$ maximum permissible speed to avoid slipping ?
On a banked road, the horizontal component of the normal force and the frictional force contribute to provide centripetal force to keep the car moving on a circular turn without slipping. At the optimum speed, the normal reaction’s component is enough to provide the needed centripetal force, and the frictional force is not needed. The optimum speed $v_o$ is given by Eq.
$v_{o}=(R g \tan \theta)^{1 / 2}$
Here $R=300 \,m , \theta=15^{\circ}, g=9.8 \,m s ^{-2} ;$ we
have $v_{o}=28.1 \,m s ^{-1}$
The maximum permissible speed $v_{\max }$ is given by Eq.
$v_{\max }=\left(R g \frac{\mu_{s}+\tan \theta}{1-\mu_{s} \tan \theta}\right)^{1 / 2}=38.1 \,m s ^{-1} $
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