$A$ particle starts from rest and performs circular motion of constant radius with speed given by $v = \alpha \sqrt{x}$,where $\alpha$ is a constant and $x$ is the distance covered. The correct graph of the magnitude of its tangential acceleration $(a_t)$ and centripetal acceleration $(a_c)$ versus $t$ will be:

$A$ particle is moving with a uniform speed in a circular orbit of radius $R$ in a central force inversely proportional to the $n^{th}$ power of $R$. If the period of rotation of the particle is $T$,then

$A$ ball of mass $m$ is attached to the free end of a string of length $l$. The ball is moving in a horizontal circular path about the vertical axis as shown in the diagram. The angular velocity $\omega$ of the ball will be ($T =$ Tension in the string).

One end of a string of length $l$ is connected to a particle of mass $m$ and the other to a small peg on a smooth horizontal table. If the particle moves in a circle with speed $v$,the net force on the particle (directed towards the centre) is :
$(i) \; T$
$(ii) \; T - \frac{m v^{2}}{l}$
$(iii) \; T + \frac{m v^{2}}{l}$
$(iv) \; 0$
$T$ is the tension in the string. [Choose the correct alternative].

$A$ train is moving towards north. At one place it turns towards north-east,here we observe that

$A$ wire of length $2.5 \ m$ is fixed at one end and a box of mass $4 \ kg$ is tied at the other end. If the wire rotates in a horizontal circle about the fixed end with $\frac{2}{\pi} \ rev/s$ rotations per second,then the tension in the wire is (in $N$)