A mass is supported on a frictionless horizontal surface. It is attached to a string and rotates about a fixed centre at an angular velocity \(\omega_0.\) If the length of the string and the angular velocity are doubled, the tension in the string, which was initially \(T_0\), is now:
1. \(T_0\)
2. \(T_0/2\)
3. \(4T_0\)
4. \(8T_0\)

Three identical particles are joined together by a thread as shown in the given figure. All three particles are moving in horizontal circles centred at \(O\). If the velocity of the outermost particle is \(v_0\), then the ratio of tensions in the three sections of the string is:

1. \(3 : 5 : 7\)
2. \(3 : 4 : 5\)
3. \(7 : 11 : 6\)
4. \(3 : 5 : 6\)

A block of mass \(m\) at the end of a string is whirled round in a vertical circle of radius \(R\). The critical speed of the block at the top of its swing below which the string would slacken before the block reaches the top is:

1. \(Rg\)
2. \((Rg)^2\)
3.  \(R/g\)
4. \(\sqrt{Rg}\)

A bucket tied at the end of a \(1.6\ \text{m}\) long string is whirled in a vertical circle with constant speed. The minimum speed when the bucket is at the highest position, so that the water from the bucket does not spill, is: (Take \(g = 10\ \text{m/s}^2\)) 
1. \(4\ \text{m/s}\)
2. \(6.25\ \text{m/s}\)
3. \(16\ \text{m/s}\)
4. None of the above

A \(1\ \text{kg}\) stone at the end of \(1\ \text{m}\) long string is whirled in a vertical circle at a constant speed of \(4\ \text{m/s}\). The tension in the string is \(6\ \text{N}\), when the stone is at: (Take \(g = 10\ \text{m/s}^2\)) 

1. Top of the circle
2. Bottom of the circle
3. Half way down
4. None of the above

The tension in the string revolving in a vertical circle with a mass \(m\) at the end, which is at the lowest position:
1. \(\dfrac{mv^2}{r}\)
2. \(\dfrac{mv^2}{r}-mg\)
3. \(\dfrac{mv^2}{r}+mg\)
4. \(mg\)

A coin, placed on a rotating turntable, slips when it is placed at a distance of \(9\ \text{cm}\) from the centre. If the angular velocity of the turntable is tripled, it will just slip if its distance from the centre is:

1. \(27\ \text{cm}\)
2. \(9\ \text{cm}\)
3. \(3\ \text{cm}\)
4. \(1\ \text{cm}\)

A body of mass \(0.4\ \text{kg}\) is whirled in a vertical circle, making \(2\ \text{rev/s}\). If the radius of the circle is \(2\ \text{m}\), then the tension in the string when the body is at the top of the circle is:

1. \(41.56\ \text{N}\)
2. \(89.86\ \text{N}\)
3. \(109.86\ \text{N}\)
4. \(122.4\ \text{N}\)

A bucket full of water is revolved in vertical circle of radius 2m. What should be the maximum time-period of revolution so that the water doesn't fall off the bucket ?

1. 1 sec

2. 2 sec

3. 3 sec

4. 4 sec

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

1. $\frac{ML{\omega }^2}{2}$

2. $ML{\omega }^2$

3. $\frac{ML{\omega }^2}{4}$

4. $\frac{M{L}^2{\omega }^2}{2}$