Does the mass of a brick increase when you lift it?

Question

If a person takes a brick, and lifts it to a height of 1 m, will the mass of the brick increase?

Answer 1

No, mass does not increase when you lift an object. Potential energy increases, but that's because of the $\Delta h$ in $U_{\mathrm{grav}} = mg\Delta h$, not the $m$. Mass is, to put it a bit imprecisely, the amount of matter in an object. When you lift something, there doesn't become more of it.

(Note that I'm ignoring relativistic effects.)

Answer 2

If you have a proton (mass=$m_p$) and a neutron (mass=$m_n$) and allow them to join via the strong nuclear force the mass of the $^2_1H$ nucleus $m_{np}$ is less than the sum of the masses of the individual particles ($m_{np} \lt m_p + m_n$).
During this joining together the proton-neutron system loses potential energy and that energy is called the binding energy of the nucleus.

The same is true of your brick-Earth system except that now the mass of the brick-Earth system when the brick has been lifted a height $h$ is higher than that of the brick-Earth system before the brick was lifted up.
Work has had to be done to lift the brick and so the brick-Earth system has gained potential energy. The change in mass of the brick-Earth system is rather small $\approx 3 \times 10^{-16}$ kg for a 3 kg brick raised by 1 metre.
Even if you took the brick a long way away from the Earth (to "infinity") the brick-Earth system would only increase in mass by approximately $2 \times 10^{-9}$ kg.

The rest mass of the brick does not change.

Answer 3

Mass certainly will change. In fact we have two connected bodies - brick and earth. The Hamiltonian (full energy operator) of a connected particle system is defined as: $H = H(brick) + H(earth) + V$, where $V$ is energy of interaction (potential energy in our case). Since potential energy will increase, mass inevitably increases due to the theory of special relativity.