Two vectors $$\overrightarrow A $$ and $$\overrightarrow B $$ are defined as $$\overrightarrow A $$ $$=$$ $$a\widehat i$$ and $$\overrightarrow B = a$$ $$\left( {\cos \,\omega T\widehat i + \sin \,\omega t\,\widehat j} \right),$$ where $$a$$ is a constant and $$\omega = \pi /6\,\,rad{s^{ - 1}}.$$ If $$\left| {\overrightarrow A + \overrightarrow B } \right| = \sqrt 3 \left| {\overrightarrow A - \overrightarrow B } \right|$$ at time $$t = \tau $$ for the first time, the value of $$\tau ,$$ in second, is ______________.

A ring and disc are initially at rest, side by side, at the top of an inclined plane which makes an angle $${60^ \circ }$$ with the horizontal. They start to roll without slipping at the same instant of time along the shortest path. If the time difference between their reaching the ground is $$\left( {2 - \sqrt 3 } \right)/\sqrt {10} \,\,s,$$ then the height of the top of the inclined plane, in metres is ______________ . Take $$g = 10\,\,m{s^{ - 2}}.$$

The densities of two solid spheres A and B of the same radii R vary with radial distance r as $${\rho _A}(r) = k\left( {{r \over R}} \right)$$ and $${\rho _B}(r) = k{\left( {{r \over R}} \right)^5}$$, , respectively, where k is a constant. The moments of inertia of the individual spheres about axes passing through their centres are $${I_A}$$ and $${I_B}$$, respectively. If, $${{{I_B}} \over {{I_A}}} = {n \over {10}}$$, the value of n is

Two identical uniform discs roll without slipping on two different surfaces AB and CD (see figure) starting at A and C with linear speeds v₁ and v₂, respectively, and always remain in contact with the surfaces. If they reach B and D with the same linear speed and v₁ = 3 m/s, then v₂ in m/s is (g = 10 m/s²)