A block of mass \(10~\text{kg}\) is in contact with the inner wall of a hollow cylindrical drum of radius \(1~\text{m}\). The coefficient of friction between the block and the inner wall of the cylinder is \(0.1\). The minimum angular velocity needed for the cylinder, which is vertical and rotating about its axis, will be: \(\left(g= 10~\text{m/s}^2\right )\)
1. \(10~\pi~\text{rad/s}\)
2. \(\sqrt{10}~\pi~\text{rad/s}\)
3. \(\frac{10}{2\pi}~\text{rad/s}\)
4. \(10~\text{rad/s}\)

Two stones of masses \(m\) and \(2m\) are whirled in horizontal circles, the heavier one in a radius \(\frac{r}{2}\) and the lighter one in a radius \(r\). The tangential speed of lighter stone is \(n\) times that of the value of heavier stone when they experience the same centripetal forces. The value of \(n\) is:

A tube of length \( L\) is filled completely with an incompressible liquid of mass \(M\) and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity \(\omega\). The force exerted by the liquid at the other end is:

A particle of mass m is tied to a string of length \(l\) and whirled into a horizontal plane. If the tension in the string is T, then the speed of the particle will be:

1. $\sqrt{\frac{Tl}{m}}$

2. $\sqrt{\frac{2Tl}{m}}$

3. $\sqrt{\frac{3Tl}{m}}$

4. $\sqrt{\frac{T}{ml}}$