Banking of Roads

The metro which has been recently introduced in Mumbai encounters a sharp turn between Andheri and Chakala. To avoid any derailing issues, the authorities thought of banking the rails. The turn is of a radius of $400 \mathrm{m}$ and the maximum speed attained by Mumbai metro is$72 \mathrm{km} {\mathrm{hr}}^{-1}$. If the distance between the rails is $1\mathrm{m}$ then through what height should the outer rail be raised?

A particle describes a horizontal circle in a conical funnel whose inner surface is smooth with the speed of $0.5 \mathrm{m} {\mathrm{s}}^{-1}.$ What is the height of the plane of the circle from the vertex of the funnel? [Take $g=10 \mathrm{m} {\mathrm{s}}^{-2}$] 

A person with his hands in his pockets is skating on ice at the velocity of $10 \mathrm{m} {\mathrm{s}}^{-1}$ and describes a circle of radius $50 \mathrm{m}.$ What is his inclination with vertical?

The radius of the curved road on the national highway is $r\mathit{.}$ The width of the road is $l$. The outer edge of the road is raised by $h$ with respect to the inner edge so that a car with velocity $v$ can pass safely over it. The value of $h$ is

A car of mass $1000 \mathrm{kg}$ negotiates a banked curve of radius $90 \mathrm{m}$ on a frictionless road. If the banking angle is $45°,$ the speed of the car is

An aircraft executes a horizontal loop with a speed of $150 \mathrm{m} {\mathrm{s}}^{-1}$ with its wings banked at an angle of $12°$. The radius of the loop is $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$
 

A cyclist goes around a circular path of circumference $34.3 \mathrm{m}$ in $\sqrt{22} \mathrm{s}$. The angle made by him, with the vertical is

A car is moving at a speed. of $60 \mathrm{km} {\mathrm{h}}^{-1}$ traversing a circular road track of radius $60 \mathrm{m}$. The minimum coefficient of friction to prevent the skidding of the car is $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

The maximum speed of a car on a road-turn of radius $30 \mathrm{m},$ if the coefficient of friction between the tyres and the road is $0.4$, will be

A body is moving in a circular orbit with static friction $0.2$. If radius through which the body revolves is $100 \mathrm{m}$ and $\mathrm{g}=9.8 \mathrm{m} {\mathrm{s}}^{-2},$ then the maximum speed with which body revolved is

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 ${\mu }_{\mathrm{s}}$. The maximum safe velocity on this road is

A car is travelling on a circular banked road. The centripetal acceleration of a car is provided by

A train is moving towards the north. At one place, it turns towards north-east. Here, we observe that

A cyclist taking turn bends inwards while a car passenger taking the same turn is thrown outwards. The reason is

A car sometimes overturns while taking a turn. When it overturns, it is

Banking of roads is independent of

The maximum speed with which a vehicle can negotiate a curved road, which is banked at an angle $\theta ={\tan }^{-1}\left(0.24\right),$ is $54 \mathrm{km} {\mathrm{hr}}^{-1}.$ If the same road is flat and the vehicle has to negotiate the curve with same maximum speed, the coefficient of friction between the road and tyres of the vehicle should be
 

A cyclist going around a circular road of radius $10 \mathrm{m}$ is observed to be bending inward $30°$ with vertical. Frictional force acting on the cyclist is (Given: $g=10 \mathrm{m} {\mathrm{s}}^{-2},$ mass of the cyclist is $90 \mathrm{kg}$ )
 

The maximum safe speed for which a banked road is intended is to be increased by $20\%.$ If the angle of banking is not changed, then the radius of curvature of the road should be changed from $30 \mathrm{m}$ to

A cyclist -with combined mass $80 \mathrm{kg}$ goes around a curved road with a uniform speed $20 \mathrm{m} {\mathrm{s}}^{-1}.$ He has to bend inward by an angle $\theta ={\tan }^{-1}\left(0.50\right)$ with the vertical. The force of friction acting at the point of contact of tyres and road surface is $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$