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Converted document POWER FOR AN INSTANTANEOUS CENTER

POWER FOR AN INSTANTANEOUS CENTER

Find the power for a rolling circular body, as seen in Figure 1↓.
figure images/InstantaneousCenter.png
Figure 1 Circular body of radius r with coordinate axes originating at center point C and an instantaneous center I.

Figure 1↑ depicts a rolling cylindrical body where the velocity of the center vC is moving along one axial direction (vC = vCi for example). To derive the kinetic energy for this example, see Kinetic Energy of an Instantaneous Center.
The rate of change of kinetic energy is equivalent to power: P = KĖ; thus power will be:
(dKE)/(dt)  =  (d)/(dt)⎛⎝(1)/(2)mvC⋅vC + (1)/(2)Ic(ωk)⋅(ωk)⎞⎠        =  (1)/(2)(mvC⋅aC)2 + (1)/(2)IC(αk⋅ωk)2        =  mac⋅vC + ICαk⋅(ωk)        =  ΣF⋅vC + ΣMC(ωk)        =  F1v1 + F2v2 + ⋯ + (M1 + M + 2 + ⋯)⋅ωk
Here, IC is the moment of inertia about point C, m is the mass of the body, and d is the distance from point C to point I. These calculations assume that the body MUST be rigid (the body does not deform or bend).