Effect of internal forces on kinetic energy
For a two-body system in absence of external forces, the kinetic energy as measured from ground frame is $K_0$ and from center of mass frame is $K_{cm}$. Find the correct statements:
$1.$ The kinetic energy as measured from center of mass frame is least.
$2.$ Only the portion of energy $K_{cm}$ can be transformed from one form to another due to internal changes in the system.
$3.$ The system always retains at least $K_0-K_{cm}$ amount of kinetic energy as measured from ground frame irrespective of any kind of internal changes in the system.
$4.$ The system always retains at least $K_{cm}$ amount of kinetic energy as measured from ground frame irrespective of any kind of internal changes in the system.
I'm confused that internal forces changes kinetic energy with respect to center of mass frame or ground frame? I found a similar question here. But it didn't address my concern.
My attempt:
Say we have a system of particles with masses $m_i$ and velocities $\vec v_i$. 512
The total kinetic energy is then given by,
$$E=\frac{1}{2}\sum m_iv_i^2$$
Let’s define a couple useful quantities: total mass, $M\equiv\sum m_i$, velocity of the center of mass, $\vec V\equiv\frac{1}{M}\sum m_i\vec v_i$. Notation for the particle velocities in the center of mass frame: $\vec u_i=\vec v_i-\vec V$
Using these, we can rewrite the kinetic energy as $\begin{align} E&=\frac{1}{2}\sum m_i(\vec V+\vec u_i)^2\ &=\frac{1}{2}\sum m_i(V^2+u_1^2+2\vec V.\vec u_i)\ &=\frac{1}{2}MV^2+\frac{1}{2}\sum m_iu_i^2+\vec V.\sum m_i.\vec u_i \end{align}$
I know that the first term is the kinetic energy from the motion of the center of mass, the second term is the kinetic energy from the motion relative to the center of mass and the third term is linear momentum of system w.r.t. the center of mass.
My question : How will the third term affect the energy of the system if it is acted by an internal force? 512