$A$ 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 $\,m/s$)

The normal reaction $N$ for a vehicle of $800 \, kg$ mass,negotiating a turn on a $30^{\circ}$ banked road at maximum possible speed without skidding is $... \times 10^{3} \, kg \cdot m/s^{2}$. [Given $\cos 30^{\circ} = 0.87, \mu_{s} = 0.2$]

$A$ motorcyclist moving with a velocity of $72\, km/h$ on a flat road takes a turn at a point where the radius of curvature of the road is $20\, m$. The acceleration due to gravity is $10\, m/s^2$. In order to avoid skidding,he must not bend with respect to the vertical plane by an angle greater than

The coefficient of friction between the tyres and the road is $0.25$. The maximum speed with which a car can be driven round a curve of radius $40 \,m$ without skidding is ........ $ms^{-1}$ (assume $g = 10 \,ms^{-2}$)

$A$ particle on the inner rough surface of a cone rotating about its vertical axis is at rest relative to the cone at a height of $1 \ m$ above its vertex. If the coefficient of friction is $\mu = 0.5$ and the semi-vertical angle of the cone is $45^\circ$,what is the maximum angular velocity $\omega$ of the cone?

$A$ motorcyclist wants to drive in horizontal circles on the vertical inner surface of a large cylindrical wooden well of radius $8.0 \ m$,with a minimum speed of $5 \sqrt{5} \ m \ s^{-1}$. The minimum value of the coefficient of friction between the tyres and the wall of the well must be (take $g = 10 \ m \ s^{-2}$):