Banking of Roads

Three particles, each of mass $m,$ are situated at the vertices of an equilateral triangle of side length $a$. The only forces acting on the particles are their mutual gravitational forces. It is desired that each particle moves in a circle while maintaining the original mutual separation $a$. Find the initial velocity that should be given to each particle and also the time period of the circular motion.

A car travels on a banked horizontal road with inclination $45°$. What is its maximum speed for not skidding up $\left(\mu =0.75,R=200\right)$.

A car is moving on a circular road of radius $150 \mathrm{m}$ with a coefficient of friction $0.6$. With what optimum velocity should the car be moved so that it crosses the turn safely.:

A body is revolving with a constant speed along a circle. If its direction of motion is reversed but the speed remains the same, then,

A string breaks if its tension exceeds $10 \mathrm{N}$. A stone of mass $250 \mathrm{g}$, tied to this string of length $10 \mathrm{cm}$, is rotated in a horizontal circle. The maximum angular velocity of rotation can be,

A car of mass $2000 \mathrm{Kg}$ is moving with a speed of 10 $\mathrm{m}{\mathrm{s}}^{-1}$on a circular path of radius $20 \mathrm{m}$ on a level road. What must be the frictional force between the car and the road so that the car does not slip?

The minimum velocity (in $\mathrm{m}{\mathrm{s}}^{-1}$) with which a car driver must traverse a flat curve of radius 150 m and coefficient of friction 0.6 to avoid skidding is

If a cyclist moving with a speed of $4.9 \mathrm{m}/\mathrm{s}$ on a level road can take a sharp circular turn of radius $4\mathrm{ }\mathrm{m}$, then coefficient of friction between the cycle tyres and road is

If a car moves on a circular banked road the centripetal force may be provided by :-

A biker initially at rest, starts to move on a circular path of radius $R=\frac{200}{7} \mathrm{km}$  with tangential acceleration $a = 1 \mathrm{m} {\sec }^{-2}$ (constant in magnitude). If on an average bike runs $50 \mathrm{km} {\mathrm{litre}}^{-1}$ and initially the bike has $2 \mathrm{litre}$ petrol. The minimum value of the coefficient of friction, so, that bike will not slip during motion is 

A stone of mass $0.03 \mathrm{kg}$ attached to a $1.5 \mathrm{m}$ long string is whirled around in a horizontal circle at a speed of $6 {\mathrm{ms}}^{-1}$. The tension in the string is:

When a car of mass $M$ passes through a convex bridge of radius $r$ with velocity $v$, then it exerts a force on it. What is the magnitude of the force?

A motor cyclist moving with a velocity of $72 \mathrm{km} {\mathrm{hour}}^{-1}$ on a flat road takes a turn on the road at a point where the radius of curvature of the road is $20 \mathrm{m}$. The acceleration due to gravity is $10 \mathrm{m} {\sec }^{-2}$. In order to avoid skidding, he must not bend with respect to the vertical plane by an angle greater than

A particle of charge-q & mass $m$ moves in a circle of radius $r$ around an infinitely long line charge on linear charge density $+\lambda .$ Then time period will be

A particle describes a horizontal circle of radius r in a funnel type vessel of frictionless surface with half one angle $\theta$ (as shown in figure). If mass of the particle is m, then in dynamical equilibrium the speed of the particle must be –

A disc revolves with a speed of $120$ $\mathrm{rev}./\min .$ and has a radius of $15 \mathrm{cm}.$ Two coins $\mathrm{A}$ and $\mathrm{B}$ are placed at $4 \mathrm{cm}$ and $14 \mathrm{cm}$ away from the centre of the record. If the coefficient of friction between the coins and the record is $0.16.$ which of the coins will revolve with the record ?