$A$ train is running at $20 \, m/s$ on a railway line with a radius of curvature of $40,000 \, m$. The distance between the two rails is $1.5 \, m$. For safe running of the train,the elevation of the outer rail over the inner rail is ...... $mm$ $(g = 10 \, m/s^2)$.

$A$ particle of mass $M$ moves with constant speed along a circular path of radius $r$ under the action of a force $F$. Its speed is

$A$ ball of mass $m$ is attached to the free end of a string of length $l$. The ball is moving in a horizontal circular path about the vertical axis as shown in the diagram. The angular velocity $\omega$ of the ball will be ($T =$ Tension in the string).

One end of a string of length $l$ is connected to a particle of mass $m$ and the other end is connected to a small peg on a smooth horizontal table. If the particle moves in a circle with speed $v$,the net force on the particle (directed towards the center) will be ($T$ represents the tension in the string).

$A$ dancer moves counterclockwise at a constant speed around the path shown below. The path is such that the lengths of its segments,$PQ, QR, RS$,and $SP$,are equal. Arcs $QR$ and $SP$ are semicircles. Which of the following best represents the magnitude of the dancer's acceleration as a function of time $t$ during one trip around the path,beginning at point $P$?

$A$ stick has its bottom end attached to a wall by a pivot and is held up by a massless string attached to its other end. Which of the following scenarios has the smallest tension in the string? (Length of stick is same in all scenarios)