Question

The friction coefficient between a road and the tyre of a vehicle is $4/3$. Find the maximum incline the road may have so that once hard brakes are applied and the wheel starts skidding, the vehicle going down at a speed of $36$ km/hr is stopped within $5$m.

Answer 1

$a=0-\frac{{10}^2}{2}\times 5$

$a=10\frac{m}{s^2}$

$R-mg\cos \theta =0$

from the free body diagram

$ma+mg\theta -\mu R=0----------------1$

$ma+mg\theta -\mu mg\theta =0$

$a+g\sin \theta -\mu g\cos \theta =0$

$10+10\times \sin \theta -\frac{4}{3}\times 10\theta =0$

$30+30\sin \theta -40\theta =0$

$3+3\sin \theta -4\theta =0$

$4\cos \theta -3\theta =3$

squaring we get

$=4\sqrt{(1-\sin ²\theta )}=9+9\sin ²\theta +18\sin \theta \sin \theta$

$=18+\sqrt{\frac{18²-4\left(25\right)\times (-7)}{2\times 25}}$

$\sin \theta =\frac{-18+32}{50}=\frac{14}{50}=0.28$

$\theta =\sin ⁻¹\left[0.28\right)={16}^{\circ }$