Page:Elementary Text-book of Physics (Anthony, 1897).djvu/49

a particle having a mass equal to the sum of all the masses if it were acted on by a force equal to the resultant of all the forces.

Forces which act between particles belonging to the same system are called internal forces; such forces do not affect the motion of the centre of mass, for, by Newton's third law of motion, they always occur in pairs, of which the two members are equal and opposite. They therefore contribute nothing to the resultant force, and so do not influence the acceleration of the centre of mass. If the only forces which act be internal forces, the acceleration of the centre of mass is zero and the momentum of the system remains constant. This principle is known as the conservation of momentum.

31. Kinetic Energy of a System of Particles.— The kinetic energy of a system of particles may also be expressed in terms of the velocity of the centre of mass. Represent by $u,v,w$ the components of velocity of each particle, by $U,V,W$ the components of velocity of the centre of mass, and by $a,b,c$ the components of velocity of each particle relative to the centre of mass. We have then

$u_{1}=U+a_{1},$$u_{2}=U+a_{2},...$
$v_{1}=V+b_{1},$$v_{2}=V+b_{2},...$
$w_{1}=W+c_{1},$$w_{2}=W+c_{2},...$

The kinetic energy of the particle $m_{1}$, is ${\tfrac {1}{2}}m_{1}(u_{1}^{2}+v_{1}^{2}+w_{1}^{2})$, and the kinetic energy of all the particles or of the system is the sum of the similar expressions obtained for each particle of the system. Substitute in the equation for the kinetic energy the values of $u^{2},v^{2},w^{2}$. We consider first the values of $u^{2}$. We have

$u_{1}^{2}=U^{2}+a_{1}^{2}+2a_{1}U,$$u_{2}^{2}=U^{2}+a_{2}^{2}+2a_{2}U,...$

Multiplying by ${\tfrac {1}{2}}m$ and adding, we obtain

$\sum {\tfrac {1}{2}}mu^{2}={\tfrac {1}{2}}U^{2}(m_{1}+m_{2}+...)+{\tfrac {1}{2}}m_{1}a_{1}^{2}+{\tfrac {1}{2}}m_{2}a_{2}^{2}+...+U(m_{1}a_{1}+m_{2}a_{2}+...).$

Now since $a_{1},a_{2},....$ are referred to the centre of mass as origin, and since in that case the coordinates of the centre of mass are zero, the sum $m_{1}a_{1}+m_{2}a_{2}...$ must equal zero. If the expres-