Why is it incorrect to say that KE of COM is not the same as total KE? Does this also apply to PE? I sort of understand why KE(com) is not the same as KE(total)... correct me if I'm wrong.. its like when a symmetric body is performing rotational motion even though the COM is not moving it still has (Rotational) kinetic energy. But does this translate to potential energy or can you calculate PE be considering the COM?
 A: For an extended body in a uniform gravitational field, its potential energy is the same as that of a point mass located at the bodies' center of mass.  This can be seen by writing out the total gravitational potential energy in terms of the gravitational potential energies of its constituent masses:
$$
U = \sum_{i} m_i g z_i = g \sum_i m_i z_i
$$
where $z$ is the "vertical" coordinate and the sum runs over all of the masses in the body (perhaps as an integral for a continuous body.)  But by the $z$-coordinate of the center of mass is
$$
z_{CM} = \frac{\sum_i m_i z_i}{M} \quad \Rightarrow \quad M z_{CM} = \sum_i m_i z_i
$$
where $M$ is the total mass of the body;  and so $U = M g z_{CM}$ for the body.
Note, however, that the same argument does not hold if the gravitational field is not uniform.  For example, you cannot say that the gravitational potential energy of a hypothetical space elevator is the same as the gravitational potential energy of a point mass at its center of mass, since the acceleration due to gravity varies substantially over the length of the object.
