$A$ particle is moving along a straight line parallel to $x-$ axis with constant velocity. Its angular momentum about the origin

$A$ particle is moving uniformly along a straight line as shown in the figure. During the motion of the particle from $A$ to $B$,the angular momentum of the particle about '$O$' is:

$A$ uniform rod of mass $M$ and length $L$ has an impulse $J$ applied at right angles to one end. If the other end begins to move with speed $V$,the magnitude of the impulse is:

$A$ particle of mass $m$ is moving in time $t$ on a trajectory given by $\overrightarrow{r} = 10 \alpha t^2 \hat{i} + 5 \beta (t - 5) \hat{j}$. Where $\alpha$ and $\beta$ are dimensional constants. The angular momentum of the particle becomes the same as it was for $t = 0$ at time $t = \dots$ seconds.

$A$ point mass $m$ is attached to one end of a string passing through a cylindrical tube. The string is held in hand,and the point mass moves in a horizontal circle of radius $r_1 = 2 \ m$ with a speed of $v_1 = 4 \ m/s$. The string is then pulled down so that the radius is reduced to $r_2 = 1 \ m$. Calculate the new linear velocity,angular velocity,and the ratio of the final kinetic energy to the initial kinetic energy.

Obtain the relation between torque of a system of particles and angular momentum.