Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass: Show $\text{K}=\text{K}'+\frac{1}{2}\text{M}\text{V}^2$ where K' is the total kinetic energy of the system of particles, K′ is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and $\frac{\text{MV}^2}{2}$ is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system). The result has been used in $\sec7.14$
Exercise
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We have the relation for velocity of the $I^{th}$ particle as, $\text{v}_\text{i}=\text{v}'_\text{i}+\text{V}$
$\sum\limits_\text{i}\text{m}_\text{i}\text{v}_\text{i}=\sum\limits_\text{i}\text{m}_\text{i}\text{v}_\text{i}'+\sum\limits_\text{i}\text{m}_\text{i}\text{V}\ ...(\text{ii})$ Taking the dot product of equation (ii) with itself, we get $\sum\limits_\text{i}\text{m}_\text{i}\text{v}_\text{i}\cdot\sum_\limits\text{i}\text{m}_\text{i}\text{v}_\text{i}=\sum\limits_\text{i}\text{m}_\text{i}\big(\text{v}'_\text{i}+\text{V}\big)\cdot\sum\limits_\text{i}\text{m}_\text{i}\big(\text{v}'_\text{i}+\text{V}\big)$
$\text{M}^2\sum\limits_\text{i}\text{v}_\text{i}^2=\text{M}^2\sum\limits_\text{i}\text{v}'{_\text{i}^2}+\text{M}^2\sum\limits_\text{i}\text{v}_\text{i}\cdot\text{v}_\text{i}'+\text{M}^2\sum\limits_\text{i} \text{v}'_\text{i}\cdot\text{v}_\text{i}+\text{M}^2\text{V}^2$ Here, for the centre of mass of the system of particles, $\sum\limits_\text{i}\text{v}_\text{i}\cdot\text{v}'_\text{i}=-\sum\limits_\text{i}\text{v}'_\text{i}\cdot\text{v}_\text{i}$
$\text{M}^2\sum\limits_\text{i}\text{v}_\text{i}^2=\text{M}^2\sum\limits_\text{i}\text{v}'{_\text{i}^2}+\text{M}^2\text{V}^2$
$\text{M}^2\sum\limits_\text{i}\text{v}_\text{i}^2=\text{M}^2\sum\limits_\text{i}\text{v}'{_\text{i}^2}+\text{M}^2\text{V}^2$
$\frac{1}{2}\text{M}\sum\limits_\text{i}\text{v}^2_\text{i}=\frac{1}{2}\text{M}\sum\limits_\text{i}\text{v}'^2_\text{i}+\frac{1}{2}\text{MV}^2$
$\text{K}=\text{K}'+\frac{1}{2}\text{MV}^2$ Where, $\text{K}=\frac{1}{2}\text{M}\sum\limits_\text{i}\text{v}^2_\text{i}$ = Total kinetic energy of the system of particles $\text{K}'=\frac{1}{2}\text{M}\sum\limits_\text{i}\text{v}'^2_\text{i}$ = Total kinetic energy of the system of particles with respect to the centre of mass $\frac{1}{2}\text{MV}^2$ = Kinetic energy of the translation of the system as a whole.
$\sum\limits_\text{i}\text{m}_\text{i}\text{v}_\text{i}=\sum\limits_\text{i}\text{m}_\text{i}\text{v}_\text{i}'+\sum\limits_\text{i}\text{m}_\text{i}\text{V}\ ...(\text{ii})$ Taking the dot product of equation (ii) with itself, we get $\sum\limits_\text{i}\text{m}_\text{i}\text{v}_\text{i}\cdot\sum_\limits\text{i}\text{m}_\text{i}\text{v}_\text{i}=\sum\limits_\text{i}\text{m}_\text{i}\big(\text{v}'_\text{i}+\text{V}\big)\cdot\sum\limits_\text{i}\text{m}_\text{i}\big(\text{v}'_\text{i}+\text{V}\big)$
$\text{M}^2\sum\limits_\text{i}\text{v}_\text{i}^2=\text{M}^2\sum\limits_\text{i}\text{v}'{_\text{i}^2}+\text{M}^2\sum\limits_\text{i}\text{v}_\text{i}\cdot\text{v}_\text{i}'+\text{M}^2\sum\limits_\text{i} \text{v}'_\text{i}\cdot\text{v}_\text{i}+\text{M}^2\text{V}^2$ Here, for the centre of mass of the system of particles, $\sum\limits_\text{i}\text{v}_\text{i}\cdot\text{v}'_\text{i}=-\sum\limits_\text{i}\text{v}'_\text{i}\cdot\text{v}_\text{i}$
$\text{M}^2\sum\limits_\text{i}\text{v}_\text{i}^2=\text{M}^2\sum\limits_\text{i}\text{v}'{_\text{i}^2}+\text{M}^2\text{V}^2$
$\text{M}^2\sum\limits_\text{i}\text{v}_\text{i}^2=\text{M}^2\sum\limits_\text{i}\text{v}'{_\text{i}^2}+\text{M}^2\text{V}^2$
$\frac{1}{2}\text{M}\sum\limits_\text{i}\text{v}^2_\text{i}=\frac{1}{2}\text{M}\sum\limits_\text{i}\text{v}'^2_\text{i}+\frac{1}{2}\text{MV}^2$
$\text{K}=\text{K}'+\frac{1}{2}\text{MV}^2$ Where, $\text{K}=\frac{1}{2}\text{M}\sum\limits_\text{i}\text{v}^2_\text{i}$ = Total kinetic energy of the system of particles $\text{K}'=\frac{1}{2}\text{M}\sum\limits_\text{i}\text{v}'^2_\text{i}$ = Total kinetic energy of the system of particles with respect to the centre of mass $\frac{1}{2}\text{MV}^2$ = Kinetic energy of the translation of the system as a whole.
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