$A$ block follows the path as shown in the figure from height $h$. If the radius of the circular path is $r$,then the relation that holds good to complete a full circle is

What is the minimum velocity at the lowest point of a vertical circle of radius $l$ such that the string just becomes slack at the highest point?

$A$ body crosses the topmost point of a vertical circle with critical speed. Its centripetal acceleration,when the string is horizontal,will be

Concrete mixture is made by mixing cement,stone,and sand in a rotating cylindrical drum. If the drum rotates too fast,the ingredients remain stuck to the wall of the drum and proper mixing of ingredients does not take place. The maximum rotational speed of the drum in revolutions per minute (rpm) to ensure proper mixing is close to (Take the radius of the drum to be $1.25\, m$ and its axle to be horizontal).

$A$ small sphere is moving at a constant speed in a vertical circle. Below is a list of quantities that could be used to describe some aspect of the motion of the sphere.
$I$ - Kinetic energy
$II$ - Gravitational potential energy
$III$ - Momentum
Which of these quantities will change as this sphere moves around the circle?

$A$ particle with charge $Q$ coulomb,tied at the end of an inextensible string of length $R$ metre,revolves in a vertical plane. At the centre of the circular trajectory,there is a fixed charge of magnitude $Q$ coulomb. The mass of the moving charge $M$ is such that $Mg = \frac{Q^2}{4 \pi \epsilon_0 R^2}$. If at the highest position of the particle,the tension of the string just vanishes,the horizontal velocity at the lowest point has to be