A point mass \(m\) is moved in a vertical circle of radius \(r\) with the help of a string. The velocity of the mass is \(\sqrt{7 g r} \) at the lowest point. The tension in the string at the lowest point will be: 
1. \(6mg\)
2. \(7mg\)
3. \(8mg\)
4. \(mg\)

A mass \(m\) is attached to a thin wire and whirled in a vertical circle. The wire is most likely to break when:

A particle of mass \(m\) having speed \(v\) goes in a vertical circular motion such that its centre is at its origin, as shown in the figure. If at any instant the angle made by the string with a negative \(y\text-\)axis is \(\theta\) then the tension in the string is:
[Take radius = \(R\)]

                    
1. \(mg\sin\theta+ \frac{mv^2}{R}\)
2. \(mg\cos\theta- \frac{mv^2}{R}\)
3. \(mg\cos\theta+ \frac{mv^2}{R}\)
4. \(mg\sin\theta- \frac{mv^2}{R}\)

A bucket full of water tied with the help of a \(2\) m long string performs a vertical circular motion. The minimum angular velocity of the bucket at the uppermost point so that water will not fall will be:
1. \(2\sqrt{5}\) rad/s
2. \(\sqrt{5}\) rad/s
3. \(5\) rad/s
4. \(10\) rad/s

The kinetic energy \(K\) of a particle moving in a circular path varies with the distance covered \(S\) as \(K = aS^2\) where \(a\) is constant. The angle between the tangential force and the net force acting on the particle is: (\(R\) is the radius of the circular path)
1. \(\tan^{-1}\left(\frac{S}{R}\right)\)
2. \(\tan^{-1}\left(\frac{R}{S}\right)\)
3. \(\tan^{-1}\left(\frac{a}{R}\right)\)
4. \(\tan^{-1}\left(\frac{R}{a}\right)\)

The angle between the position vector and the acceleration vector of a particle in a non-uniform circular motion (centre of the circle is taken as the origin) will be:
1. \(0^\circ\)
2. \(45^\circ\)
3. \(75^\circ\)
4. \(135^\circ\)