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 $v_{\mathrm{o}}$ as shown in the figure. If the maximum height attained by an object is $\frac{7 v_0^2}{10 \mathrm{~g}}$ ( $\mathrm{g}=$ acceleration due to gravity), the object is a $\_\_\_\_$ .

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 $2 R$. 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 $I_2 / I_1=$ $\_\_\_\_$

The position of an object having mass 0.1 kg as a function of time $t$ is given as

$\vec{r} = \left( 10 t^2 \hat{i} + 5 t^3 \hat{j} \right)$ m. At $t = 1$ s, which of the following statements are correct?

A. The linear momentum $\vec{p} = \left( 2 \hat{i} + 1.5 \hat{j} \right)$ kg·m/s.

B. The force acting on the object $\vec{F} = \left( 2 \hat{i} + 3 \hat{j} \right)$ N.

C. The angular momentum of the object about its origin $\vec{L} = 15 \hat{k}$ J·s.

D. The torque acting on the object about its origin $\vec{\tau} = 20 \hat{k}$ N·m.

Choose the correct answer from the options given below: