If $\vec{\omega} = (3\hat{i} + 4\hat{j} + 5\hat{k}) \text{ rad/s}$ and $\vec{r} = (\hat{i} - 2\hat{j} + 3\hat{k}) \text{ m}$,then the linear velocity vector of the particle is:

$A$ ladder of length $L$ is slipping with its ends against a vertical wall and a horizontal floor. At a certain moment,the speed of the end in contact with the horizontal floor is $v$ and the ladder makes an angle $\alpha = 30^o$ with the horizontal. If $dv/dt = 0$,then the angular acceleration of the ladder when $\alpha = 45^o$ is:

In pure translational motion,the velocity of every particle of a body at any instant is what? Equal or unequal?

$A$ rod of length $L$ is held vertically on a smooth horizontal surface. The top end of the rod is given a gentle push. At a certain instant of time,when the rod makes an angle $\theta$ with the horizontal,the velocity of the $COM$ of the rod is $v_0$. The velocity of the end of the rod in contact with the surface at that instant is:

All particles of a body move in circular paths when its axis of rotation.........

The mass of the pulley is $m$ and its radius is $R$. Assume the pulley to be a uniform disc. The pulley is free to rotate about an axis passing through its centre and perpendicular to its plane. The string is massless and inextensible,and it is wrapped on the pulley. There is no slipping between the string and the pulley. The length of the string which is not wrapped on the pulley is $2R$. $A$ block of mass $m$ is released from the position as shown in the figure. The impulse exerted by the string on the pulley at the moment the string becomes taut is $J$. The value of $\frac{2m \sqrt{gR}}{J}$ is equal to: