If the equation for the displacement of a particle moving on a circular path is given by:

                               $\theta  = 2t^3 + 0.5$

Where $\theta $ is in radian and $t$ in second, then the angular velocity of the particle at $t = 2\,sec$ is $t=$ ....... $rad/sec$

$A$ particle of mass $m$ is projected with a velocity $u$ making an angle $45^o$ with the horizontal. The magnitude of the torque due to weight of the projectile, when the particle is at its maximum height, about the point of projectile

Two particles of equal masses have velocities$\overrightarrow {{v_1}}  = 2\hat i\,m/s$ and $\overrightarrow {{v_2}}  = 2\hat j\,m/s$. The first particle has an acceleration $\overrightarrow {{a_1}}  = \left( {3i + 3\hat j} \right)\,m/{s^2}$,while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a

Four masses are fixed on a massless rod as shown in Fig. The moment of inertia about the axis $P$ is about ....... $kg-m^2$

A circular disk of moment of inertia $I_t$ is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed $\omega _i$. Another disk of moment of inertia $I_b$ is dropped coaxially onto the rotating disk. Initially the second disk has zero angular speed. Eventually both the disks rotate with a constant angular speed $\omega _f$. The energy lost by the initially rotating disc to friction is

A child is standing at one end of a long trolley moving with a speed  $v$  on a smooth horizontal floor. If the child starts running towards the other end of the trolley with a speed $u,$  the centre of mass of the system (trolley + child) will move with a speed.