#title: Sistemas de N partículas

#author: Ricardo

#let:

n m[3]  = {[1..9]}; 

n x0[3] = {[-9..-1,1..9]}; 

n y0[3] = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, -1, -2, -3, -4, -5, -6, -7, -8, -9]; 

n u0[3] = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, -1, -2, -3, -4, -5, -6, -7, -8, -9]; 

n v0[3] = [ 1, 2, 3, 4, 5, 6, 7, 8, 9, -1, -2, -3, -4, -5, -6, -7, -8, -9]; 

s q[4] = [posição do centro de massa,
          velocidade do cento de massa,
          momento linear total,
          momento angular total relativamente à origem,
          energia cinética total,
          energia cinética  do centro de massa
           ] ;

n r1[4] = [(#m[0]*#x0[0]+#m[1]*#x0[1]+#m[2]*#x0[2])/(#m[0]+#m[1]+#m[2]),
          (#m[0]*#u0[0]+#m[1]*#u0[1]+#m[2]*#u0[2])/(#m[0]+#m[1]+#m[2]),
          (#m[0]*#u0[0]+#m[1]*#u0[1]+#m[2]*#u0[2]),
           (#m[0]*(#x0[0]*#v0[0]-#y0[0]*#u0[0])+#m[1]*(#x0[1]*#v0[1]-#y0[1]*#u0[1])+#m[2]*(#x0[2]*#v0[2]-#y0[2]*#u0[2])),
           (#m[0]*(#u0[0]*#u0[0]+#v0[0]*#v0[0])+#m[1]*(#u0[1]*#u0[1]+#v0[1]*#v0[1])+#m[2]*(#u0[2]*#u0[2]+#v0[2]*#v0[2]))/2,
           ((#m[0]*#u0[0]+#m[1]*#u0[1]+#m[2]*#u0[2])*(#m[0]*#u0[0]+#m[1]*#u0[1]+#m[2]*#u0[2])/(#m[0]+#m[1]+#m[2]))/2
           ] ;

n r2[4] = [(#m[0]*#y0[0]+#m[1]*#y0[1]+#m[2]*#y0[2])/(#m[0]+#m[1]+#m[2]),
           (#m[0]*#v0[0]+#m[1]*#v0[1]+#m[2]*#v0[2])/(#m[0]+#m[1]+#m[2]),
            (#m[0]*#v0[0]+#m[1]*#v0[1]+#m[2]*#v0[2]),
            ~,
            ~,
            ~ ] ;

s sol1[4] = [ $x=\frac{\sum_{i=1}^3 x_i m_i}{\sum_{i=1}^3m_i}=$, 
              $v_x=\frac{\sum_{i=1}^3 v_{xi} m_i}{\sum_{i=1}^3m_i}=$, 
              $p_x=\sum_{i=1}^3 v_{xi} m_i=$, 
              $L_z=\sum_{i=1}^3 m_i(x_iv_{yi}-y_iv_{xi})=$,
              $T=\frac{1}{2}\sum_{i=1}^3 m_i(v_{xi}^2+v_{yi}^2)=$,
              $T_c=\frac{1}{2}\left(\sum_{i=1}^3m_i\right)\left(\frac{\sum_{i=1}^3 v_{xi} m_i}{\sum_{i=1}^3m_i}\right)^2=$
               ] ;

s sol2[4] = [ $y=\frac{\sum_{i=1}^3 y_{i} m_i}{\sum_{i=1}^3m_i}=$, 
               $v_y=\frac{\sum_{i=1}^3 v_{yi} m_i}{\sum_{i=1}^3m_i}=$, 
               $p_y=\sum_{i=1}^3 v_{yi} m_i=$, 
               ~,
            ~,
            ~ ] ;



q ~ r1 ~r2 ~ sol1 ~ sol2 ;         


#question:

Considere 3 partículas com as seguintes propriedades, respectivamente:\\
~\\
massas (em kg): $m_1=#m[0]$, $m_2=#m[1]$, $m_3=#m[2]$; \\
~\\
coordenadas (em m): $(#x0[0],#y0[0])$, $(#x0[1],#y0[1])$, $(#x0[2],#y0[2])$;   \\
~\\
velocidades (em m/s):  $(#u0[0],#v0[0])$, $(#u0[1],#v0[1])$, $(#u0[2],#v0[2])$. 

\vspace{5mm}

Calcule:
\begin{enumerate}

\item #q[0];
\item #q[1];
\item #q[2];
\item #q[3];

\end{enumerate}

#sugestion:
    
#resolution:
  
\begin{enumerate}

\item #sol1[0]#r1[0] \hspace{5mm}  #sol2[0]#r2[0];
\item #sol1[1]#r1[1] \hspace{5mm}  #sol2[1]#r2[1];
\item #sol1[2]#r1[2] \hspace{5mm}  #sol2[2]#r2[2];
\item #sol1[3]#r1[3] \hspace{5mm}  #sol2[3]#r2[3];

\end{enumerate}


#result:

\begin{enumerate}

\item #r1[0] \hspace{5mm}  #r2[0];
\item #r1[1] \hspace{5mm}  #r2[1];
\item #r1[2] \hspace{5mm}  #r2[2];
\item #r1[3] \hspace{5mm}  #r2[3];

\end{enumerate}


    
#Verify:

#Obs: