Analogy to binding energy

Can I understand the relationship between binding energy and mass by comparing it to the relationship between kinetic energy and potential energy?

When an object falls, its gravitational potential, a scalar value, decreases, and its energy is converted into kinetic energy. Even when a nuclear fusion reaction occurs, the mass, which is a scalar value, decreases, and it is converted into binding energy and released. The relationship between the two seems similar. Can I understand the relationship between binding energy and mass by comparing it to the relationship between kinetic energy and potential energy?

No, potential energy and kinetic energy are just two forms of energy. Mass is not a form of energy. Despite what is commonly said, you cannot convert mass to energy. The products of a nuclear reaction, including all particles and radiation, have the same combined mass as everything before the reaction occurred.

When we say that mass is converted to energy in a reaction, we mean that the non-radiation particles end up with less mass and we usually ignore the fact that the energy contained in the form of the kinetic energy of the particles and the energy carried in the released radiation all has mass. In other words, the invariant mass of the particles is different before and after the reaction, but the total mass of the system is conserved. Invariant mass is commonly called 'rest mass'.

If we look at a nucleus in a high energy state that decays and releases a gamma ray, the nucleus has more mass prior to the decay and less mass afterwards. But that gamma ray has energy, and thus it has to have mass. It just doesn't have invariant mass since a photon can't be brought to rest. If that gamma ray then excites a nearby nucleus into a high-energy state, that mass is transferred from the first nucleus to the second, through the gamma ray photon.

hongseok said:

Even when a nuclear fusion reaction occurs, the mass, which is a scalar value, decreases, and it is converted into binding energy and released.

Personally I would not say that mass is 'converted' to anything here. Two protons and two neutrons have more mass when kept apart than when they are brought together and allowed to bind into an alpha particle. But that binding process releases a lot of energy, and the mass of that energy is equal to the difference between the masses of the particles before and after binding. The mass didn't disappear, it just move from the invariant mass of the particles to the kinetic energy and any radiation emitted.

Drakkith said:

Mass is not a form of energy. Despite what is commonly said, you cannot convert mass to energy.

I disagree with this. Mass is rest energy.

The point is that ”energy” by itself is not a mysterious substance of any sort. Nothing can be converted ”to energy” but different forms of energy can be converted to each other - always keeping the total the same.

Drakkith said:

The products of a nuclear reaction, including all particles and radiation, have the same combined mass as everything before the reaction occurred.

This is misleading. The full system has the same invariant mass, but the sum of the masses of the constituents certainly changes.

Drakkith said:

the energy contained in the form of the kinetic energy of the particles and the energy carried in the released radiation all has mass

Photons certainly do not have mass. Although they can contribute to the invariant mass of a system.

Drakkith said:

But that gamma ray has energy, and thus it has to have mass.

It does not have mass. The only mass in relativity is the invariant mass and a gamma ray has zero invariant mass. However, it gives a contribution to the invariant mass of the system.

Orodruin said:

I disagree with this. Mass is rest energy.

The point is that ”energy” by itself is not a mysterious substance of any sort. Nothing can be converted ”to energy” but different forms of energy can be converted to each other - always keeping the total the same.

I don't quite agree that mass is rest energy, but I'm not going to quibble about it since I'm not a physicist. My point was that I wouldn't say that mass is converted into energy, so the analogy wasn't really valid.

Orodruin said:

This is misleading. The full system has the same invariant mass, but the sum of the masses of the constituents certainly changes.

This was what I was trying to get across. If we account for everything before and after the reaction the mass of the system is the same. It's just that the mass has been moved around a bit.

Orodruin said:

Photons certainly do not have mass. Although they can contribute to the invariant mass of a system.

My apologies, I worded my response poorly. Photons do not have mass, but their energy contributes mass to the system of which they are a part.

Orodruin said:

It does not have mass. The only mass in relativity is the invariant mass and a gamma ray has zero invariant mass. However, it gives a contribution to the invariant mass of the system.

Like I said, I worded my response poorly.

Drakkith said:

I don't quite agree that mass is rest energy, but I'm not going to quibble about it since I'm not a physicist.

In relativity, rest energy is how we define mass. One of the great insights of special relativity is that this mass is actually the same thing as the inertial mass in the rest frame of an object that we are familiar with from Newtonian mechanics - therefore justifying calling the rest energy "mass".

I will graciously accept your explanation and bow to your expertise. As much as my bad back and knees allow, that is.