Show that a system of particles moving under the influence of an external force, moves as if the force is applied at its centre of mass.

Explanation

Consider a system of particles of masses. `m_1, m_2, ......m_n` moves with velocity vectors `vecv_1,vec v_2, vecv_3...... , vecv_n`
`therefore` Velocity of centre of mass
`vecV_(CM) = 1/M {m_1vecv_1 + m_2 vecv_2 + .....+ m_nvecv_n}`
`because m_1 +m_2 + ......+m_n = M`
We know
` therefore veca = (d)/(dt) (vecV)`
`therefore` Acceleration of centre of mass
`a_(CM) = (d)/(dt)(vecV_(CM))`
`therefore a_(CM) = 1/M {m_1 (d)/(dt) vecv_1 +m_2 (d)/(dt)vecv_2 +......+m_n(d)/(dt)vecv_n}`
But Force (F) = ma, so total force on the body is
`F = Ma_(CM) = m_1 a_1 + m_2a_2 +m_3 a_3 + ........+ m_na_n`
or Total Force F ` = ma_(CM) = F_1 +F_2 +F_3 +........+F_n`
Hence, total force on the body is the sum of forces on individual particles and it is equals to force on centre of mass of the body.