Rotational Motion

A solid sphere (A) of mass $5m$ and a spherical shell (B) of mass $m$, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of A and B, they start rolling without slipping with an acceleration of $a_A$ and $a_B$, respectively. The ratio of $a_A$ and $a_B$ is ______.

A solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The smaller part having mass $M/8$ is converted into a sphere of radius $r$ and the larger part is converted into a circular disc of thickness $t$ and radius $2R$. If $I_1$ is moment of inertia of a sphere having radius $r$ about an axis through its centre and $I_2$ is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia $I_2/I_1=$ ______.

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first 2 s it rotates through an angle $\theta_1$ and in the next 2 s it rotates through an angle $\theta_2$. The ratio $\frac{\theta_2}{\theta_1}$ is ______.

An object of uniform density rolls up the curved path with the initial velocity $v_0$ as shown in the figure. If the maximum height attained by an object is $\frac{7v_0^2}{10g}$ ($g$ = acceleration due to gravity), the object is a ______.

A solid sphere of radius 4 cm and mass 5 kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of 1200 rpm. It is brought to rest in 10 s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are ______ and ________ respectively.

A solid cylinder having radius $R$ and length $L$ is slipping on a rough horizontal plane. At time $t=0$ the cylinder has a translational velocity $v_0=49$ m/s, perpendicular to its axis and a rotational velocity $v_0/4R$ about the centre. The time taken by the cylinder to start rolling is ______ seconds. (coefficient of kinetic friction $\mu_K=0.25$ and $g=9.8$ m/s$^2$)

The position vectors of two 1 kg particles, (A) and (B), are given by $\vec r_A=(\alpha_1t^2\hat i+\alpha_2t\hat j+\alpha_3t\hat k)$ m and $\vec r_B=(\beta_1t\hat i+\beta_2t^2\hat j+\beta_3t\hat k)$ m, respectively; ($\alpha_1=1$ m/s$^2$, $\alpha_2=3n$ m/s, $\alpha_3=2$ m/s, $\beta_1=2$ m/s, $\beta_2=-1$ m/s$^2$, $\beta_3=4p$ m/s), where t is time, n and p are constants. At $t=1$ s, $|\vec V_A|=|\vec V_B|$ and velocities $\vec V_A$ and $\vec V_B$ of the particles are orthogonal to each other. At $t=1$ s, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt{L}$ kg m$^2$s$^{-1}$. The value of L is ______.

The torque due to the force $(2\hat{i} + \hat{j} + 2\hat{k})$ about the origin, acting on a particle whose position vector is $(\hat{i} + \hat{j} + \hat{k})$, would be

A tube of length 1m is filled completely with an ideal liquid of mass 2M, and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is F then angular velocity of the tube is $\sqrt{\dfrac{F}{\alpha M}}$ in SI unit. The value of $\alpha$ is ______.