Systems of Particles

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Section 21.2 Systems of Particles

Consider a system of \(n\) different masses \(m_i\text{,}\) interacting with each other and being acted on by external forces. We can write Newton’s second law for the positions \(\rr_i\) of each of these masses with respect to a fixed origin \(\OOb\text{,}\) thereby obtaining a system of equations governing the motion of the masses.

\begin{align} m_1 \frac{d^2\rr_1}{dt^2} \amp= \FF_1 + 0 + \ff_{12} + \ff_{13} + . . . + \ff_{1n}\notag\\ m_2 \frac{d^2\rr_2}{dt^2} \amp= \FF_2 + \ff_{21} + 0 + \ff_{23} + . . . + \ff_{2n}\tag{21.2.1}\\ \vdots \amp\notag\\ m_n \frac{d^2\rr_n}{dt^2} \amp= \FF_n + \ff_{n1} + \ff_{n2} + . . . + \ff_{n(n-1)} + 0\notag \end{align}

Here, we have chosen the notation \(\FF_i\) for the net external forces acting on mass \(m_i\) and \(\ff_{ij}\) for the internal force of mass \(m_j\) acting on \(m_i\text{.}\)

Figure 21.2. The position vectors \(\rr_1\) and \(\rr_2\) for masses \(m_1\) and \(m_2\) and the displacement vector \(\rr=\rr_2-\rr_1\) between them.

Figure 21.2 shows the basic geometry for the special case of only two masses. \(\rr_1\) and \(\rr_2\) are the position vectors of the two masses measured with respect to an arbitrary coordinate origin \(\OOb\text{.}\) We call the displacement between the two masses \(\rr\text{,}\) the magnitude of this displacement \(r\text{,}\) and the direction \(\rhat\text{.}\) These quantities can be found from \(\rr_1\) and \(\rr_2\) by

\begin{align} \rr\amp= \rr_2 -\rr_1\tag{21.2.2}\\ r \amp= |\rr| = |\rr_2-\rr_1|\tag{21.2.3}\\ \rhat \amp= \frac{\rr}{r}\text{.}\tag{21.2.4} \end{align}

In this interactive figure, you can drag the masses to different positions in all three dimensions. Notice that while \(\rr_1\) and \(\rr_2\) depend on the origin of coordinates, the relative displacement vector \(\rr=\rr_2-\rr_1\) does not depend on the choice of origin.

In general, each internal force \(\ff_{ij}\) will depend on the positions of the particles \(\rr_i\) and \(\rr_j\) in some complicated way, making (21.2.1) a set of coupled differential equations. To solve the system (21.2.1), we first need to decouple the differential equations, i.e. find an equivalent set of differential equations in which each equation contains only one variable.