Systems of Particles

The mass for a system of particles is the sum of the masses of each particle:

\begin{align} m = \sum m_i \end{align}

The position vector for a system of particles is the weighted average of position vectors by mass, or the center of mass:

\begin{align} \mathbf{r}_c = \frac{\sum m_i\mathbf{r}_i}{\sum m_i} \end{align}

The velocity vector is the velocity of the center of mass:

\begin{align} \mathbf{v}_c = \frac{\sum m_i\mathbf{v}_i}{\sum m_i} \end{align}

The acceleration vector is the acceleration of the center of mass:

\begin{align} \mathbf{a}_c = \frac{\sum m_i\mathbf{a}_i}{\sum m_i} \end{align}

The linear momentum of a system of particles is the sum of the linear momenta of its constituents:

\begin{align} \mathbf{G} = m\mathbf{v}_c \end{align}

The angular momentum of a system of particles is the sum of the angular momenta of its constituents:

\begin{align} \mathbf{H}^P = \sum(\mathbf{r}_i - \mathbf{r}_p)\times m_i\mathbf{v}_i = \mathbf{H}^C + (\mathbf{r}_C-\mathbf{r}_P)\times G \end{align}

The kinetic energy of a system of particles is the sum of all its individual kinetic energies:

\begin{align} T = \sum \frac{1}{2}m_i\mathbf{v}_i \cdot \mathbf{v}_i = \frac{1}{2}m\mathbf{v}_C\cdot\mathbf{v}_C + \frac{1}{2}\sum m_i(\mathbf{v}_i-\mathbf{v}_C)\cdot(\mathbf{v}_i-\mathbf{v}_C) \end{align}

The balance of linear momentum for a system of particles sums the balance of linear momentum for each particle:

\begin{align} \mathbf{F} = m\mathbf{a}_C \end{align}

The balance of angular momentum for a system of particles is derived by taking the derivative of angular momentum for a system of particles:

\begin{align} \dot{\mathbf{H}}^P = \mathbf{M}^P - \mathbf{v}_P\times\mathbf{G} \end{align}

Specifically, if the point \(P\) is the center of mass \(C\), we can simplify this to:

\begin{align} \dot{\mathbf{H}}^C = \mathbf{M}^C \end{align}

Additionally, if \(P\) is some fixed origin \(O\) (with \(\mathbf{v}_O = 0\)), we can simplify to:

\begin{align} \dot{\mathbf{H}}^O = \mathbf{M}^O \end{align}

Taking the integral with respect to time yields the linear impulse-linear momentum equations and the angular impulse-angular momentum equations:

\begin{align} \int_{t_1}^{t_2} \mathbf{F}\text{ d}t &= \mathbf{G}_C(t_2) - \mathbf{G}_C(t_1) \\ \int_{t_1}^{t_2} \mathbf{M}^C\text{ d}t &= \mathbf{H}^C(t_2) - \mathbf{H}^C(t_1) \\ \int_{t_1}^{t_2} \mathbf{M}^O\text{ d}t &= \mathbf{H}^O(t_2) - \mathbf{H}^O(t_1) \end{align}

Finally, the work-energy theorem is the sum for each particle:

\begin{align} \boxed{T(t_2) - T(t_1) = \sum W_{\mathbf{F}_i, 12}} \end{align}