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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=Ni=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.)

clipboard_ec06fcb9e098b7c1241a898efb4d1fafa.png
Figure 21.1 Torque diagram for center of mass reference frame

We can now apply rS,i=rS,cm+rcm,i to Equation (21.3.1) yielding

τextS=Ni=1(rS,i×Fi)=Ni=1((rcm,i+rcm,i)×Fi)=Ni=1(rS,cm×Fi)+Ni=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).

clipboard_e9852af9e32b4e3234cf8ae0b9525852a.png
Figure 21.2 Torque diagram for “point-like” particle located at center of mass

The term,

τextcm=Ni=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=Ni=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+Ni=1(rcm,i×mvcm,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=Ni=1(rcm,i×mvcm,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=dLsysSdt

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=dLotbial Sdt+dLspinSdt

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

dLobital Sdt=ddt(rS,cm×psys)

We apply the vector identity

ddt(A×B)=dAdt×B+A×dBdt

to Equation (21.3.13) yielding

dLorbital Sdt=drS,cmdt×psys+rS,cm×dpsysdt

The first term in Equation (21.3.21) is zero because

drS,cmdt×psys=Vcm×mtotalVcm=0

Therefore the time derivative of the orbital angular momentum about a point S, Equation (21.3.15), becomes

dLorbital Sdt=rS,cm×dpsysdt

In Equation (21.3.17), the time derivative of the momentum of the system is the external force,

Fext=dpsysdt

The expression in Equation (21.3.17) then becomes the first of our relations

dLorbital 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),

dLsinSdt=ddtNi=1(rcm,i×mivcm,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

dLspinSdt=Ni=1(drcm,idt×mivcm,i)+Ni=1(rcm,i×ddt(mivcm,i))

The first term in Equation (21.3.21) is zero because

Ni=1(drcm,idt×mivcm,i)=Ni=1(vcm,i×mivcm,i)=0

Therefore the time derivative of the spin angular momentum about the center of mass, Equation (21.3.21), becomes

dLspinSdt=Ni=1(rcm,i×ddt(mivcm,i))

The force, acting on an element of mass mi, is

Fi=ddt(mivcm,i)

The expression in Equation (21.3.23) then becomes

dLspinSdt=Ni=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

dLsinSdt=Ni=1(rcm,i×Fexti)=Ni=1τextcm,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=dLotbial 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=Ni=1(rcm,i×Fexti)=dLspincmdt


This page titled 21.3: Translational and Rotational Equations of Motion is shared under a not declared license and was authored, remixed, and/or curated by Peter Dourmashkin (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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