Physics with Calculus/Mechanics/Center of Mass

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Let r_i be the vector from the origin to the ith particle. Let m_i, v_i, f_i be the mass, velocity, and force of the ith particle respectively. In order to see if there are any particularly special points, define an arbitrary vector R. Now, every  r_i can be written as  r_i' + R. Applying Newton's second law to every particle,

 f_i = m_i \frac{d v_i}{dt}.


 v_i = \frac{d r_i'}{dt} + \frac{dR}{dt} = v_i' + V.

(the last equation is just definition). So

 f_i = m_i \frac{d v_i'}{dt} + m_i \frac{dV}{dt} .


 F = \sum_i f_i = \frac{d}{dt} \sum_i  m_i v_i' + M \frac{dV}{dt}.


 \sum_i m_i \frac{dr_i'}{dt} = P.

Where P is a constant. Thus,

 \sum_i m_i r_i' = Pt + C

where C is a constant. Substituting the definition of R,

 \sum_i m_i R - \sum m_i r_i = Pt + C .

This is equivalent to

 R = \frac{1}{\sum_i m_i} \sum_i m_i r_i + \frac{1}{\sum_i m_i} Pt + \frac{1}{\sum_i m_i} C .



 R = \frac{1}{\sum_i m_i} \sum_i m_i r_i.


 F = M \frac{d^2 R}{dt^2}


 F = \sum_i f_i


 M = \sum_i m_i.



\sum f_i = \sum_i f_i^{external} + \sum_{i,j} f_{ij}

