A mass of $10\mathrm{ }\mathrm{kg}$ is suspended vertically by a rope from the roof. When a horizontal force is applied on the rope at some point, the rope deviated at an angle of $45°$ at the roof point. If the suspended mass is at equilibrium, the magnitude of the force applied is $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

A smooth wire is bent into a vertical circle of a radius $a$. A bead $P$ can slide smoothly on the wire. The circle is rotated about the vertical diameter $AB$ as the axis with speed $\omega$ as shown in the figure. The bead $P$ is at rest with respect to the circular ring in the position shown. Then, ${\omega }^2$ is equal to,

A disc of radius $R$ has a light pole fixed perpendicular to the disc at the circumference which in turn has a pendulum of length $R$ attached to its other end as shown in the figure. The disc is rotated with a constant angular velocity $\omega$. The string makes an angle $30°$ with the rod. Then, the angular velocity of the disc is,

Two different masses are connected to two strings, as shown in the figure. Both masses rotate about a central fixed point with angular velocity $10 \mathrm{rad} {\mathrm{s}}^{-1}$, on a smooth horizontal plane. The ratio of tensions $\left(\frac{T_1}{T_2}\right)$ in the strings is