$A$ particle moves on a circular overbridge with constant speed. The friction coefficient varies so that the speed remains constant. Which of the following graphs shows the magnitude of friction $f$ versus the angle $\theta$?

$A$ car of $800 \,kg$ is taking a turn on a banked road of radius $300 \,m$ and angle of banking $30^{\circ}$. If the coefficient of static friction is $0.2$, then the maximum speed with which the car can negotiate the turn safely is: $(g=10 \,m/s^2, \sqrt{3}=1.73)$ (in $\,m/s$)

$A$ man is standing on a rough $(\mu = 0.5)$ horizontal disc rotating with a constant angular velocity of $5 \ rad/s$. At what distance from the center should he stand so that he does not slip on the disc?

Two cars are moving on a banked circular path of radius $8 \ m$ having an angle of banking $45^{\circ}$. If the coefficients of static friction between the road and the tyres of the two cars are $0.5$ and $0.4$ respectively,then the ratio of maximum permissible speeds of the cars to avoid slipping is

$A$ train runs along an unbanked circular track of radius $30 \; m$ at a speed of $54 \; km/h$. The mass of the train is $10^{6} \; kg$. What provides the centripetal force required for this purpose - the engine or the rails? What is the angle of banking required to prevent wearing out of the rail?

$A$ car is moving on a horizontal circular road of radius $0.1 \, km$ with constant speed. If the coefficient of friction between the tyres of the car and the road is $0.4$,then the speed of the car may be ......... $m/s$ $(g = 10 \, m/s^2)$.