A car is negotiating a curved road of radius $R$. The road is banked at an angle $\theta .$ The coefficient of friction between the tyres of the car and the road is $\mu _s.$ The maximum safe velocity on this road is
NEET 2016, Difficult
Download our app for free and get started
For vertical equilibrium on the road,
$N \cos \theta=m g+f \sin \theta$
$m g=N \cos \theta-f \sin \theta$
Centripetal force for safeturning,
$N \sin \theta+f \cos \theta=\frac{m v^2}{R}$
From eqns. $(i)$ and $(ii)$, we get
$\frac{v^2}{R g}=\frac{N \sin \theta+f \cos \theta}{N \cos \theta-f \sin \theta}$
$\Rightarrow \frac{v_{\max }^2}{R g} =\frac{N \sin \theta+\mu_s N \cos \theta}{N \cos \theta-\mu_s N \sin \theta}$
$v_{\max } =\sqrt{R g\left(\frac{\mu_s+\tan \theta}{1-\mu_s \tan \theta}\right)}$
$N \cos \theta=m g+f \sin \theta$
$m g=N \cos \theta-f \sin \theta$
Centripetal force for safeturning,
$N \sin \theta+f \cos \theta=\frac{m v^2}{R}$
From eqns. $(i)$ and $(ii)$, we get
$\frac{v^2}{R g}=\frac{N \sin \theta+f \cos \theta}{N \cos \theta-f \sin \theta}$
$\Rightarrow \frac{v_{\max }^2}{R g} =\frac{N \sin \theta+\mu_s N \cos \theta}{N \cos \theta-\mu_s N \sin \theta}$
$v_{\max } =\sqrt{R g\left(\frac{\mu_s+\tan \theta}{1-\mu_s \tan \theta}\right)}$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1A car is moving on a horizontal circular track of radius $0.2 \,km$ with a constant speed. If coefficient of friction between tyres of car and road is $0.45$, then speed of car may be ........ $m / s$ [Take $g=10 \,m / s ^2$ ]View Solution
- 2A lift is moving downwards with an acceleration equal to acceleration due to gravity. $A$ body of mass $M$ kept on the floor of the lift is pulled horizontally. If the coefficient of friction is $\mu $, then the frictional resistance offered by the body isView Solution
- 3Two blocks $( m =0.5\, kg$ and $M =4.5\, kg$ ) are arranged on a horizontal frictionless table as shown in figure. The coefficient of static friction between the two blocks is $\frac{3}{7} .$ Then the maximum horizontal force that can be applied on the larger block so that the blocks move together is ......... $N.$ (Round off to the Nearest Integer) [Take g as $9.8\, ms ^{-2}$ ]View Solution
- 4An isolated rail car originally moving with speed $v_0$ on a straight, frictionles, level track contains a large amount of sand. $A$ release valve on the bottom of the car malfunctions, and sand begins to pour out straight down relative to the rail car. What happens to the speed of the rail car as the sand pours out?View Solution
- 5The coefficient of static friction, $\mu _s$ between block $A$ of mass $2\,kg$ and the table as shown in the figure is $0.2$. What would be the maximum mass value of block $B$ so that the two blocks $B$ so that the two blocks do not move? The string and the pulley are assumed to be smooth and masseless ....... $kg$ $(g = 10\,m/s^2)$View Solution
- 6A body is moving along a rough horizontal surface with an initial velocity $6\,\,m/s.$ If the body comes to rest after travelling $9\, m$, then the coefficient of sliding friction will beView Solution
- 7A block of mass $10\, kg$ is kept on a rough inclined plane as shown in the figure. A force of $3\, N$ is applied on the block. The coefficient of static friction between the plane and the block is $0.6$. What should be the minimum value of force $P$, such that the block does not move downward? (take $g = 10\, ms^{-2}$) ........ $N$View Solution
- 8A block of mass $4\, kg$ rests on an inclined plane. The inclination of the plane is gradually increased. it is found that when the inclination is $3$ in $5\left( {\sin \theta = \frac{3}{5}} \right)$, the block just begins to slide down the plane. The coefficient of friction between the block and the plane isView Solution
- 9View SolutionA block pressed against the vertical wall is in equilibrium. The minimum coefficient of friction is:-
- 10For a body moving in a circular path, a condition for no skidding if $\mu $ is the coefficient of friction, isView Solution