$A$ circular road of radius $1000\, m$ has a banking angle of $45^o$. The maximum safe speed of a car having a mass of $2000\, kg$ will be,if the coefficient of friction between the tyre and the road is $0.5$.

$A$ particle is moving along the circle $x^2 + y^2 = a^2$ in an anticlockwise direction. The $x-y$ plane is a rough horizontal stationary surface. At the point $(a \cos \theta, a \sin \theta)$,the unit vector in the direction of friction on the particle is:

$A$ small object placed on a rotating horizontal turntable just slips when it is placed at a distance $4\, cm$ from the axis of rotation. If the angular velocity of the turntable is doubled,the object slips when its distance from the axis of rotation is (in $, cm$)

$A$ car is moving on a circular level road of curvature $300\,m.$ If the coefficient of friction is $0.3$ and acceleration due to gravity is $10\,m/s^2,$ the maximum speed the car can have is ........ $km/hr.$

$A$ motor car has a width of $1.1 \ m$ between its wheels. Its center of gravity is $0.62 \ m$ above the ground and the coefficient of friction between the wheels and the road is $0.8$. What is the maximum possible speed in $m/s$ if the center of gravity moves in a circle of radius $15 \ m$ on a horizontal road surface?

$A$ $70 \; kg$ man stands in contact against the inner wall of a hollow cylindrical drum of radius $3 \; m$ rotating about its vertical axis. The coefficient of friction between the wall and his clothing is $0.15$. What is the minimum rotational speed (in $rad/s$) of the cylinder to enable the man to remain stuck to the wall (without falling) when the floor is suddenly removed? (Take $g = 10 \; m/s^2$)