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There is rest mass ##m_0## (the amount of stuff an object is made off, an invariant) and the inertial mass ##m_{inertial}##.

As an object'a speed ##v## increases, its ##KE## increases, and the object gains mass ##m##" in the amount $$m=\frac {KE}{c^2}$$, correct? Is that mass increase m the inertia mass? Or do we call the entire mass (including both rest mass and extra gained mass) the

*inertial relativistic*mass?

The same goes if an object gains gravitational potential energy ##PE##: its mass increases by an amount $$m=\frac{PE}{c^2}$$

correct?

I hear that many don't like to use the term "relativistic mass". Why? Is the relativistic mass the sum of the rest mass and the gained or lost by adding energy (either potential or gravitational) to the system?

Thanks!