In the arrangement shown the fixed pulley is massless and frictionless. If the acceleration of block of mass $1 \mathrm{kg}$ is $\frac{30}{x} \mathrm{m} {\mathrm{s}}^{-2}$. Find the value of $x$.

In the given arrangement the block is released from the position where the spring is unstretched. The speed of the block when it has descended through $2\mathrm{cm}$ is $\frac{x}{10}$. Find the value of $x$.

A lamina is made by removing a small disc of diameter $2R$ from a bigger disc of uniform mass density and radius $2R\text{,}$ as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I_{\mathrm{O}}$ and $I_{\mathrm{P}}$, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\frac{I_{\mathrm{P}}}{I_{\mathrm{O}}}$ to the nearest integer is

From a uniform disk of radius $R$, an equilateral triangle of side $\sqrt{3}R$ is removed. The position of centre of mass now is $\left(0,N\right)$, then value of $N$ is ____ unit. (Given $R=12$ units)

Linear mass density of rod $AB$ varies as $\lambda =\left(x+2\right) \mathrm{kg} {\mathrm{m}}^{-1}$. Position of centre of mass from end $B$ is $\frac{N}{7}$, then value of $N$ is _____ $\mathrm{m}$.

Velocities of $2 \mathrm{kg}$ and $1 \mathrm{kg}$ block are shown at any instance. And other instant, when there would be maximum elongation in the spring, the velocity of $1 \mathrm{kg}$ block will be _____ $\mathrm{m} {\mathrm{s}}^{-1}$.

$Y$-coordinates of centre of mass of uniform plate is $\frac{1}{6}\sqrt{N}\mathrm{m}$. Find $N \_\_\_ .$

A uniform cylinder of radius $R=1.0 \mathrm{m}$ was rotated with angular velocity ${\omega }_0$ about its axis when it was placed into a corner between the horizontal floor and vertical wall, each having coefficient of friction $\mu =1$ with the cylinder. If the cylinder comes to rest in $2 \mathrm{s}$, the value of ${\omega }_0$ in $\mathrm{rad} {\mathrm{s}}^{-1}$ is ____. $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$