# Does a ticking watch have more mass? [duplicate]

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In the video on YouTube, The Real Meaning of E=mc² | PBS| Space Time Studios, it claims that a ticking watch has more mass then a non ticking watch due to the intrinsic KE, PE and thermal energy of the watch's internal movements that manifests itself as part of the watch's mass.

Is this correct ?

To me there is something wrong with this argument, but I cannot put my finger on it.

## marked as duplicate by Kyle Kanos, John Rennie, Martin, ACuriousMind♦, HDE 226868Jul 28 '15 at 16:16

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## 4 Answers

It's not so much that it is ticking that is crucial here but the fact that the watch is in a higher energy state that in its, say unwound state. This fact increases the rest mass of the watch by an amount $\Delta E/c^2$, where $\Delta E$ is the potential energy input to elastically stress the spring and thus wind the watch.

A wound, broken, unticking watch (as long as its spring can store potential energy) also has more rest mass - by the analogous amount - as an unwound, broken unticking watch.

What's wrong with it is that they didn't define the two watches very clearly. It's obvious to assume they meant identical watches (my watch is more massive than my wife's, ticking or not) but they don't specify the conditions of the watches. Really though, we only need one watch and an arbitrarily accurate scale to show what's going on.

Let's say our watch starts with a fully charged but disconnected battery. At this point it is not ticking. We weigh the watch to determine its non-ticking mass. Then we connect the battery and weigh the watch while it's ticking. Aside from small vibrations due to the clockwork, the scale will not read a higher weight/mass; in fact, the reading will decline slowly as the watch runs, until the battery "dies" and no more charge is flowing. The watch is once again not ticking, but now it is less massive than before.

The reason the mass does not increase when the battery is connected is because the kinetic energy of the clockwork, which must be included in a full inventory of the mass-energy of the system, is generated from the chemical potential energy stored in the full battery, which must also be included. When the battery runs the clockwork, that kinetic energy is then converted to heat, which is lost to the environment, reducing the overall mass-energy of the watch.

Basically what it comes down to is that a charged battery is more massive than a depleted identical battery due to the stored energy in the chemical potential, so really the main mistake was unclear writing: technically they did mention the "intrinsic potential energy" as a factor, though that only implies a charged battery without being explicit about why a charged vs. uncharged battery matters.

It is correct to say that we measure the mass of moving objects to be greater than an object at rest relative to our reference frame. Another way of saying moving object is- an object with kinetic energy.

Think about it this way, if moving objects didn't appear to gain mass then as you give them more kinetic energy they could accelerate to the speed of light. By having an increase in mass with velocity, objects become difficult to accelerate indefinitely and ultimately impossible toward the speed of light.

Does a ticking watch have more mass?

Yep. As does a wound-up watch, or a hot watch, or a watch that's higher up in a gravitational field.

It claims that a ticking watch has more mass then a non ticking watch due to the intrinsic KE, PE and thermal energy of the watch's internal movements that manifests itself as part of the watch's mass. Is this correct ?

Yep, it's what E=mc² is all about.

To me there is something wrong with this argument, but I cannot put my finger on it.

It's the enormity of it. Check out Compton scattering. That's where E=hf photon wave energy is converted into the motion of electrons:

Image courtest of Rod Nave's hyperphysics

There's nothing unusual about this, in that the photon wave energy is kinetic energy, and the motion of the electrons is kinetic energy too. In theory you could perform another Compton scatter on the residual photon until all the photon energy is converted into electron motion and there's no wave left. It has been entirely converted into electron kinetic energy.

Where things get interesting is when you realise that you could have performed pair production using that photon. To create an electron and a positron. That's when you realise that the electron is made out of kinetic energy. It's made out of the same thing as the motion of electrons. This is what Einstein's E=mc² is all about. And it's all down to the wave nature of matter. We can diffract electrons, just like we can diffract light. And electrons have spin, as evidenced by the Einstein-de Haas effect which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics". Once it clicks it's pretty obvious that the electron is akin to a photon going round and round. And pretty obvious that whilst photon momentum is resistance to change-in-motion for a wave moving linearly at c, electron mass is resistance to change-in-motion for a wave going round and round at c. Hence a radiating body loses mass. And in the case of electron-positron annihilation, it loses all of it.