A particle tied at one end of an inextensible string of length $R$ whose another end is fixed. A particle is performing circular motion in vertical plane. The ratio of maximum to minimum speed is $\lambda$. Choose the correct option(s).

$A$ rod $AB$ of length $2 \mathrm{m}$ is hinged at point $A$ and its other end $B$ is attached to a platform on which a block of mass $m$ is kept. Rod rotates about point $A$ maintaining angle $\mathrm{\theta }=30°$ with the vertical in such a way that platform remains horizontal and revolves on the horizontal circular path. If the coefficient of static friction between the block and platform is $\mu =0.1$, then find the maximum angular velocity in $\mathrm{rad} {\mathrm{s}}^{-1}$ of rod so that the block does not slip on the platform, $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

A particle of mass $5 \mathrm{kg}$ is free to slide on a smooth ring of radius $r=20 \mathrm{cm}$ fixed in a vertical plane. The particle is attached to one end of a spring whose other end is fixed to the top point $O$ of the ring. Initially, the particle is at rest at a point $A$ of the ring such that $\angle OCA=60°\text{,} C$ being the centre of the ring. The natural length of the spring is also equal to $r=20 \mathrm{cm}$. After the particle is released and slides down the ring, the contact force between the particle and the ring becomes zero when it reaches the lowest position $B$. The force constant is given as $x\times {10}^2 \mathrm{N} {\mathrm{m}}^{-1}$

Find the value of $x$

Two different masses are connected to two light and inextensible strings as shown in the figure. Both masses rotate about a central fixed point with constant angular speed of $10 \mathrm{rad} {\mathrm{s}}^{-1}$ on a smooth horizontal plane. Find the ratio of tensions $\frac{T_1}{T_2}$ in the strings. (Given, $M_1=0.25 \mathrm{kg}\text{,} M_2=1.0 \mathrm{kg}\text{,} R_1=5 \mathrm{cm}\text{,} R_2=10 \mathrm{cm}$)

Two strings of lengths, $l=0.5 \mathrm{m}$ each are connected to a block of mass, $m=2 \mathrm{kg}$ at one end and their ends are attached to the point $A$ and $B\text{,} 0.5 \mathrm{m}$ apart on a vertical pole which rotates with a constant angular velocity, $\omega =7 \mathrm{rad} {\mathrm{s}}^{-1}$. Find the ratio $\frac{T_1}{T_2}$ of tension in the upper string $\left(T_1\right)$ and the lower string $\left(T_2\right)$. [Use, $g=9.8 \mathrm{m} {\mathrm{s}}^{-2}$.]

A wedge is placed on a smooth horizontal surface. It contains a circular quadrant of radius $25 \mathrm{cm}$ as shown. An insect crawls on the circular part with a constant speed $0.5 \mathrm{m} {\mathrm{s}}^{-1}$. A force $F$ is applied on the wedge so that it does not move. Find the value of $F$ in newton when radial line of position of insect makes an angle of $60°$ with horizontal.

A small bead can slide without friction on a circular hoop that is in a vertical plane and has a radius of $10 \mathrm{m}$. The hoop rotates at a constant rate about a vertical diameter. Find the height of the bead (in $\mathrm{m}$) above the bottommost point if the angular velocity of rotation is $0.5 \mathrm{rad} {\mathrm{s}}^{-1}$.

A toy car is moving anticlockwise on a closed track whose curved portions are semicircles of radius $1 \mathrm{m}$. The adjacent graph describes the variation of speed of the car with distance moved by it (starting from point $S$). Find the time $\left(t\right)$ required for the car to complete one lap in second, (take   $\pi \ln \left(2\right)\approx 2$)