Question

Show that the total linear momentum of a
system of particles is equal to the product of the total mass of the system and the velocityof its centre of mass.​

Answer 1

Consider the expression for the position of the center of mass $\bf{\overline{x}}$ of the system consisting of $n$ particles having $m_i$ masses each situated at $\bf{x_i}$ distances from the axis of rotation, for $i=1,\ 2,\ 3,\dots,\ n.$

$\displaystyle\overline{\bf{x}}=\dfrac {\displaystyle\sum_{i=1}^nm_i\bf{x_i}}{\displaystyle\sum_{i=1}^nm_i}\\\\\\\sum_{i=1}^nm_i\bf{x_i}=\overline{\bf{x}}\displaystyle\sum_{i=1}^nm_i$

Differentiating both sides of the equation with respect to time,

$\displaystyle\dfrac {d}{dt}\left (\sum_{i=1}^nm_i\bf{x_i}\right)=\dfrac {d}{dt}\left (\overline{\bf{x}}\sum_{i=1}^nm_i\right)\\\\\\\sum_{i=1}^n\dfrac {d}{dt}\left (m_i\bf{x_i}\right)=\dfrac {d}{dt}\left (\overline{\bf{x}}\right)\sum_{i=1}^nm_i\\\\\\\sum_{i=1}^nm_i\dfrac {d}{dt}\left (\bf{x_i}\right)=\dfrac {d}{dt}\left (\overline{\bf{x}}\right)\sum_{i=1}^nm_i\\\\\\\sum_{i=1}^nm_i\bf{v_i}=\overline{\bf{v}}\displaystyle\sum_{i=1}^nm_i$

The final equation implies the total linear momentum of the system is the product of total mass of the system and the velocity of the center of mass, $\bf{\overline {v}}.$

Hence the Proof!