Torque and Angular Momentum

Separation of Motion of a system of particles into motion of the center of mass and motion about the center of mass:

Show $\frac{d{L}^{\text{'}}}{dt}=\Sigma r_i^{\text{'}}\times \frac{d{p}^{\text{'}}}{dt}$
Further, show that
$\frac{d{L}^{\text{'}}}{dt}={\tau }_{ext}^{\text{'}}$
where ${\tau }_{ext}^{\text{'}}$ is the sum of all external torques acting on the system about the centre of mass. (Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:

Show $L=L^{\text{'}}+R\times MV$
where $L^{\text{'}}=\Sigma r_i^{\text{'}}\times p_i^{\text{'}}$ is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember $r_i^{\text{'}}=r_i-R$, $r_i$ is the position of $i^{th}$ particle with respect to origin and $R$ and $V$ is the position and velocity of centre of mass with respect to origin, respectively.

A meter stick is balanced on a knife edge at its centre. When two coins, each of mass $5 \mathrm{g}$ are put one on top of the other at the $12.0 \mathrm{cm}$ mark, the stick is found to be balanced at $45.0 \mathrm{cm}$. What is the mass of the metre stick?

A child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of $40$ rev/min. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$ times the initial value? Assume that the turntable rotates without friction. Show that the child’s new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?

Two particles, each of mass $m$ and speed $v$, travel in opposite directions along parallel lines separated by a distance $d$. Show that the angular momentum vector of the two-particle system is the same whatever be the point about which the angular momentum is taken.

Find the components along the $x,y,z$ axes of the angular momentum $l$ of a particle, whose position vector is $r$ with components $x,y,z$ and momentum is $p$ with components $p_x$, $p_y$ and $p_z$. Show that if the particle moves only in the $x$- $y$ plane the angular momentum has only a $z$-component.