A particle of mass ' $m$ ' performs uniform circular motion of radius ' $r$ ' with linear speed ' $v$ ' under the application of force ' $F$ '. If ' $m$ ', ' $v$ ' and $' \mathrm{r}$ ' are all increased by $20 \%$ the necessary change in force required to maintain the particle in uniform circular motion, is

A particle rotates in a horizontal circle of radius 'R' in a conical funnel with constant speed 'V'. The inner surface of the funnel is smooth. The height of the plane of the circle from the vertex of the funnel is (g-acceleration due to gravity)

For a particle in uniform circular motion

A disc at rest is subjected to a uniform angular acceleration about its axis. Let $\theta$ and $\theta_1$ be the angle made by the disc in $2^{\text {nd }}$ and $3^{\text {rd }}$ second of its motion. The ratio $\frac{\theta}{\theta_1}$ is