The velocity and acceleration vectors of a particle undergoing circular motion are, $\overrightarrow{v}=2\hat{\mathrm{i}} \mathrm{m} {\mathrm{s}}^{-1}$ and $\overrightarrow{a}=2\hat{\mathrm{i}}+4\hat{\mathrm{j}} \mathrm{m} {\mathrm{s}}^{-2}$, respectively, at an instant of time. The radius of the circle is,

A particle $A$ moves along a circle of radius $R=50 \mathrm{cm}$ so that its radius vector $r$ relative to the fixed point $O$ (figure) rotates with the constant angular velocity $\mathrm{\omega }=0.40 \mathrm{rad} {\mathrm{s}}^{-1}$. Then modulus $v$ of the velocity of the particle, and the modulus $a$ of its total acceleration will be

A large mass $M$ is stationary at the end of a light string that passes through a smooth fixed ring to a small mass $m$ that moves around in a horizontal circular path. If $l$ is the length of the string from $m$ to the top end of the tube and $\theta$ is angle between this part and vertical part of the string as shown in the figure, then the time taken by $m$ to complete one circle is equal to,

Three identical particles are joined together by a thread as shown in figure. All the three particles are moving on a smooth horizontal plane about point $O$. If the speed of the outermost particle is ${\mathrm{v}}_0$, then the ratio of tensions in the three sections of the string is (assume that the string remains straight),

A small frictionless block slides with velocity $0.5\sqrt{gr}$ on a horizontal surface as shown in the figure. The block leaves the surface at point $C$. The angle $\theta$ in the figure is,