# Mass + potential kinetic energy

1. May 27, 2009

### thornza

Hi,

I was busy studying for my first year physics exam that's coming up and had a thought on the whole mass = energy thing.

If I carry a mass up a large hill, will it gain mass at the top of the hill due to the potential energy that has been increased?

sorry if this is a stupid question...

later,

thornza

2. May 27, 2009

### Staff: Mentor

I'd say no. The potential energy is not a property of the object, but of the object-earth system. If you consider the object-earth system as a whole, then increasing its internal energy by lifting the mass to the top of a hill will increase the mass of the system.

3. May 27, 2009

### Irrational

but is the potential energy that's gained not equal to the kinetic energy used to get it there, if looking at the object-earth (presume that includes the person who got it there) system?

4. May 27, 2009

### Staff: Mentor

I agree completely.
This is also correct, but only if the force doing the lifting is external to the object-earth system. If it is internal then there is just a transfer of energy from one form (eg chemical) of potential energy to gravitational potential energy. Such an internal transfer would result in no net change in the mass of the earth-object system.

Last edited: May 27, 2009
5. May 27, 2009

### Staff: Mentor

As DaleSpam explained, the mass of the object-earth system will increase only if the person providing the input is considered to be outside of the system. If you include the person as part of the system, then there will be no net change in mass. The person will use up chemical energy in raising the object (thus reducing his mass) while the rest of the system gains energy (and thus mass).

6. May 27, 2009

### ZikZak

Strongly disagree. The person's chemical energy did in fact come from outside the system: in particular, the Sun. Normally, that solar radiation would have been returned to space as heat, but in this case is stored on Earth as potential energy. A distant observer would indeed measure the mass (= energy in the Earth's rest frame) of the Earth/ball system to have increased.

This additional energy is not stored locally in the ball. Or in the Earth. The ball will not weigh more at the top of the hill. In fact, it will weigh less as measured by a spring . The energy is stored in the gravitational field; it will only be detected when putting a distant test mass in orbit around the Earth/ball system.

7. May 27, 2009

### Staff: Mentor

That energy entered the person-Earth-object system (including everything else on the Earth) when the person's food was growing, which was probably months ago. The exact amount of time depends of course on the particular food he eats. On the time scale of the person carrying an object up a hill, this is surely irrrelevant.

8. May 27, 2009

### ZikZak

We're talking about the extra mass involved in pushing a ball up a hill, and it's the *timescale* that's irrelevant??

It makes no difference how long that particular energy has been on Earth. It is now stored in such a way that it can no longer be thermally radiated to space. Earth/ball on hill in thermal equilibrium with the solar radiation contains more energy than Earth/ball in valley in thermal equilibrium with the solar radiation.

Last edited: May 27, 2009
9. May 27, 2009

### Staff: Mentor

:rofl: excellent point! Very funny.

I don't think there is a substantive disagreement here, just a small disagreement on the boundaries of the system. You implicitly exclude the person in the description of the system while jtbell and Doc Al explicitly include him. Earth/ball and exhausted man on hill in thermal equilibrium with the solar radiation contains the same energy as Earth/ball and energized man in valley in thermal equilibrium with the solar radiation. So, if the man (and the chemical energy he contains) is considered part of the system then the energy and mass is unchanged. Otherwise the man is an external force which does work on the system and the system's energy and mass increases.

Last edited: May 27, 2009