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 $\sqrt{7gr}$ at the lowest point. The tension in the string at the lowest point will be:

1. 6 mg

2. 7 mg

3. 8 mg

4. 1 mg

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

1. inclined at a 60$°$ angle from vertical

2. the mass is at the highest point

2. the wire is horizontal

4. the mass is at the lowest point.

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-axis is $\theta$ then the tension in the string is:
[Take radius = R]

$1.$ $mg\sin \theta +\frac{m{v}^2}{R}$
$2.\hspace{0.17em}mg\cos \theta -\frac{m{v}^2}{R}$
$3.\hspace{0.17em}\hspace{0.17em}mg\cos \theta +\frac{m{v}^2}{R}$
$4.\hspace{0.17em}\hspace{0.17em}mg\sin \theta -\frac{m{v}^2}{R}$

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 = a${\mathrm{S}}^2$, 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.  ${\tan }^{-1}\left(\frac{\mathrm{S}}{\mathrm{R}}\right)$

2.  ${\tan }^{-1}\left(\frac{\mathrm{R}}{\mathrm{S}}\right)$

3.  ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{R}}\right)$

4.  ${\tan }^{-1}\left(\frac{\mathrm{R}}{\mathrm{a}}\right)$

A stone of mass \(\mathrm{m}\) tied to the end of a string revolves in a vertical circle of radius \(\mathrm{R}.\) The net forces at the lowest and highest points of the circle directed vertically downwards are:

(${\mathrm{T}}_1$ and ${\mathrm{v}}_1$ denote the tension and speed at the lowest point.${\mathrm{T}}_2$ and ${\mathrm{v}}_2$ denote corresponding values at the highest point.)

The angle between the position vector and the acceleration vector of a particle in non-uniform circular motion (centre of the circle is taken as the origin) will be:

1. 0°

2. 45°

3. 75°

4. 135°