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KINETIC ENERGY

figure images/arbitrarybody.png
Figure 1 Arbitrary body with coordinate axes originating at point C and mass dm for an incremental volume used for integration.

Kinetic energy is associated with the motion of a body or particle and relies on the velocity of the body. The definition of translational kinetic energy (KE) for a three-dimensional body moving in two dimensions (planar motion), as seen in Figure 1↑, is as follows:
(1) KE = (1)/(2)mv2 = (1)/(2)( VV)dm
The velocity vector V at an arbitrary point (x,y) is a combination of the velocity at point C and the velocity due to rotation about the z axis on point C. Thus, the velocity is defined as:
(2) V  =  VC + ωk × (xi + yj)  =  VC + ω( − yi + xj)
where VC is the velocity vector at point C (the origin of the selected axes) and ωk is the angular velocity with respect to the z axis (i.e. the body is rotating in the x-y plane). The kinetic energy is a positive scalar value (not a vector) that is calculated using the dot product of velocity vectors.
Substituting this dot product into Equation 1↑ results into the following expression:
(3) KE = (1)/(2) VCVCdm + (ω2)/(2)(x2 + y2)dm − ω(VCi)ydm + ω(VCj)xdm
Since the following integrals are zero:
(4) ydm = xdm = 0
kinetic energy simplifies to the following equation:
(5) KE = (1)/(2)m(VCVC) + (1)/(2)ICzzω2
This expression has a component relating to translational motion, (1 ⁄ 2)m(VCVC), and another to rotational motion, (1 ⁄ 2)ICzzω2. Again, note that the kinetic energy is a scalar value, and has the units of a force times a distance, such as ftlb or Joules (Nm).