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Suppose I have some sort of rigid body, a solid sphere lets say. For simplicity's sake lets assume that the sphere can only rotate about a single arbitrary axis through the center of mass. If the center of mass of the sphere is travelling with a constant velocity (with respect to some arbitrary reference frame) and the sphere also has a non-zero constant angular velocity about a single axis, will the total kinetic energy of the sphere be equal to the sum of the Rotational and Translational kinetic energy? Intuitively, I feel like the answer should be yes, but when I try to verify my intuition, the math seems to suggest otherwise.

E

In the reference frame of the center of mass, the velocity of any particle is defined by its angular velocity and the distance from center of mass.

v

If the center of mass if moving with velocity v with respect to the reference frame we are observing the sphere at then we can apply a simple Galilean transformation to find an expression for v

v = ae

v

v

If I plug this back into the expression for the total kinetic energy, I get something odd.

E

E

I'm not sure if I didn't properly define translational and rotational energy in this case, but the above equation seems to imply that the total kinetic energy is more than the sum of the translational and rotational kinetic energy. I'm a little confused about mechanics concerned rotation right now, so any guidance would be greatly appreciated.

EDIT: Change "sphere" to flat disk, I wanted to work in 2 dimensions for simplicity's sake

E

_{total}=1/2 ∑m_{i}v^{2}_{i}In the reference frame of the center of mass, the velocity of any particle is defined by its angular velocity and the distance from center of mass.

v

^{'}_{i}= rw(cosΘ)e_{y}- rw(sinΘ)e_{x}If the center of mass if moving with velocity v with respect to the reference frame we are observing the sphere at then we can apply a simple Galilean transformation to find an expression for v

_{i}v = ae

_{y}+ be_{x}v

_{i}= (rw(cosΘ) + a)e_{y}+ (b - rw(sinΘ))e_{x}v

^{2}_{i}= (v^{'}_{i})^{2}+ v^{2}+2v^{'}_{i}⋅vIf I plug this back into the expression for the total kinetic energy, I get something odd.

E

_{total}= 1/2∑m_{i}(v^{'}_{i})^{2}+ 1/2∑m_{i}v^{2}+ ∑m_{i}v^{'}_{i}⋅vE

_{total}= E_{rotational}+ E_{translational}+ ∑m_{i}v^{'}_{i}⋅vI'm not sure if I didn't properly define translational and rotational energy in this case, but the above equation seems to imply that the total kinetic energy is more than the sum of the translational and rotational kinetic energy. I'm a little confused about mechanics concerned rotation right now, so any guidance would be greatly appreciated.

EDIT: Change "sphere" to flat disk, I wanted to work in 2 dimensions for simplicity's sake

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