21.3: Translational and Rotational Equations of Motion
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For a system of particles, the torque about a point S can be written as
→τextS=N∑i=1(→ri×→Fi)
where we have assumed that all internal torques cancel in pairs. Let choose the point S to be the origin of the reference frame O , then →rS,cm=→Rcm (Figure 21.1). (You may want to recall the main properties of the center of mass reference frame by reviewing Chapter 15.2.1.)

We can now apply →rS,i=→rS,cm+→rcm,i to Equation (21.3.1) yielding
→τextS=N∑i=1(→rS,i×→Fi)=N∑i=1((→rcm,i+→rcm,i)×→Fi)=N∑i=1(→rS,cm×→Fi)+N∑i=1(→rcm,i×→Fi)
The term
→τextS,cm=→rS,cm×→Fext
in Equation (21.3.2) corresponds to the external torque about the point S where all the external forces act at the center of mass (Figure 21.2).

The term,
→τextcm=N∑i=1(→rcm,i×→Fi)
is the sum of the torques on the individual particles in the center of mass reference frame. If we assume that all internal torques cancel in pairs, then
→τextcm=N∑i=1(→rcm,i×→Fexti)
We conclude that the external torque about the point S can be decomposed into two pieces,
→τextS=→τextS,cm+→τextcm
We showed in Chapter 20.3 that
→LsysS=→rS,cm×→psys+N∑i=1(→rcm,i×m→vcm,i)
where the first term in Equation (21.3.7) is the orbital angular momentum of the center of mass about the point S
→Lorbital S=→rS,cm×→psys
and the second term in Equation (21.3.7) is the spin angular momentum about the center of mass (independent of the point S)
→LspinS=N∑i=1(→rcm,i×m→vcm,i)
The angular momentum about the point S can therefore be decomposed into two terms
→LsysS=→LorbitalS+→LspinS
Recall that that we have previously shown that it is always true that
→τextS=d→LsysSdt
Therefore we can therefore substitute Equation (21.3.6) on the LHS of Equation (21.3.11) and substitute Equation (21.3.10) on the RHS of Equation (21.3.11) yielding as
→τextS,cm+→τextcm=d→Lotbial Sdt+d→LspinSdt
We shall now show that Equation (21.3.12) can also be decomposed in two separate conditions. We begin by analyzing the first term on the RHS of Equation (21.3.12). We differentiate Equation (21.3.8) and find that
d→Lobital Sdt=ddt(→rS,cm×→psys)
We apply the vector identity
ddt(→A×→B)=d→Adt×→B+→A×d→Bdt
to Equation (21.3.13) yielding
d→Lorbital Sdt=d→rS,cmdt×→psys+→rS,cm×d→psysdt
The first term in Equation (21.3.21) is zero because
d→rS,cmdt×→psys=→Vcm×mtotal→Vcm=→0
Therefore the time derivative of the orbital angular momentum about a point S, Equation (21.3.15), becomes
d→Lorbital Sdt=→rS,cm×d→psysdt
In Equation (21.3.17), the time derivative of the momentum of the system is the external force,
→Fext=d→psysdt
The expression in Equation (21.3.17) then becomes the first of our relations
d→Lorbital Sdt=→rS,cm×→Fext=→τextS,cm
Thus the time derivative of the orbital angular momentum about the point S is equal to the external torque about the point S where all the external forces act at the center of mass, (we treat the system as a point-like particle located at the center of mass).
We now consider the second term on the RHS of Equation (21.3.12), the time derivative of the spin angular momentum about the center of mass. We differentiate Equation (21.3.9),
d→LsinSdt=ddtN∑i=1(→rcm,i×mi→vcm,i)
We again use the product rule for taking the time derivatives of a vector product (Equation (21.3.14)). Then Equation (21.3.20) the becomes
d→LspinSdt=N∑i=1(drcm,idt×mi→vcm,i)+N∑i=1(→rcm,i×ddt(mi→vcm,i))
The first term in Equation (21.3.21) is zero because
N∑i=1(d→rcm,idt×mi→vcm,i)=N∑i=1(→vcm,i×mi→vcm,i)=→0
Therefore the time derivative of the spin angular momentum about the center of mass, Equation (21.3.21), becomes
d→LspinSdt=N∑i=1(→rcm,i×ddt(mi→vcm,i))
The force, acting on an element of mass mi, is
→Fi=ddt(mi→vcm,i)
The expression in Equation (21.3.23) then becomes
d→LspinSdt=N∑i=1(→rcm,i×→Fi)
The term, ∑Ni=1(→rcm,i×→Fi) is the sum of the torques on the individual particles in the center of mass reference frame. If we again assume that all internal torques cancel in pairs, Equation (21.3.25) may be expressed as
d→LsinSdt=N∑i=1(→rcm,i×→Fexti)=N∑i=1→τextcm,i=→τextcm
which is the second of our two relations.
Summary
For a system of particles, there are two conditions that always hold (Equations (21.3.19) and (21.3.26)) when we calculate the torque about a point S ; we treat the system as a point-like particle located at the center of mass of the system. All the external forces →Fext act at the center of mass. We calculate the orbital angular momentum of the center of mass and determine its time derivative and then apply
→τextS,cm=→rS,cm×→Fext=d→Lotbial Sdt
In addition, we calculate the torque about the center of mass due to all the forces acting on the particles in the center of mass reference frame. We calculate the time derivative of the angular momentum of the system with respect to the center of mass in the center of mass reference frame and then apply
→τextcm=N∑i=1(→rcm,i×→Fexti)=d→Lspincmdt