# Rest mass and energy

## Main Question or Discussion Point

So from what I've heard, the rest mass of a object increases when the object's internal energy increases.

So a clock that is ticking has increased rest mass vs the same clock that is completely still.

But doesn't the clock have moving parts that give kinetic and thermal energy? Then in what sense can we even talk about the clock's rest mass being increased if the ticking clock is not truly at rest?

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phinds
Gold Member
2019 Award
So from what I've heard, the rest mass of a object increases when the object's internal energy increases.

So a clock that is ticking has increased rest mass vs the same clock that is completely still.

But doesn't the clock have moving parts that give kinetic and thermal energy? Then in what sense can we even talk about the clock's rest mass being increased if the ticking clock is not truly at rest?
That's a good question and I'm not sure about the clock (because I don't know how to analyze the results due to the tradeoff between the energy bound in the spring vs the heat energy that results from the motion when a tick occurs) , but just think of a block of iron at room temperature. Heat it up to 200 degrees F and it weights more (probably not enough more to be measured by any existing instruments but still ... )

jbriggs444
Homework Helper
2019 Award
So a clock that is ticking has increased rest mass vs the same clock that is completely still.

But doesn't the clock have moving parts that give kinetic and thermal energy? Then in what sense can we even talk about the clock's rest mass being increased if the ticking clock is not truly at rest?
The rest mass (also known as "invariant mass" or just plain old "mass") of an object is equal to its total energy in the frame of reference where its total momentum is zero. [Divided by c2 if needed to make the units come out right].

So yes, if the clock has moving parts (for instance, the hour hand, the minute hand, the second hand, the flywheel, and escapement) then the kinetic energy of those parts in the clock's rest frame counts toward the clock's mass. But, of course, the mainspring is also winding down as this kinetic energy is drained into vibration and heat and eventually radiated away. So the clock's mass is decreasing. When you wind it up, its mass increases again.

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Khashishi
Well, clocks are (mostly) all digital these days, so you have a battery which is losing mass as it undergoes a chemical reaction. The mass associated with chemical bonds is stored in the electromagnetic field.

Dale
Mentor
Then in what sense can we even talk about the clock's rest mass being increased if the ticking clock is not truly at rest?
That is why I prefer the term "invariant mass" over the term "rest mass"

The rest mass (also known as "invariant mass" or just plain old "mass") of an object is equal to its total energy in the frame of reference where its total momentum is zero. [Divided by c2 if needed to make the units come out right].

So yes, if the clock has moving parts (for instance, the hour hand, the minute hand, the second hand, the flywheel, and escapement) then the kinetic energy of those parts in the clock's rest frame counts toward the clock's mass. But, of course, the mainspring is also winding down as this kinetic energy is drained into vibration and heat and eventually radiated away. So the clock's mass is decreasing. When you wind it up, its mass increases again.
Interesting. So in the object's frame of reference we know that the center of mass of the object has 0 momentum. But why does the total momentum need to be 0? Did you mean the center of mass?

Khashishi
The center of mass is a point. The object has 0 momentum, not the center of mass.
The total momentum doesn't have to be 0. In which case, you can calculate the invariant mass using
$m = \sqrt{\frac{E^2}{c^4} - \frac{p^2}{c^2}}$
By going into the frame where momentum is 0, it's simpler: $m = \frac{E}{c^2}$
But you get the same result regardless of frame. (That's why it's called "invariant")

Dale
Mentor
Interesting. So in the object's frame of reference we know that the center of mass of the object has 0 momentum. But why does the total momentum need to be 0? Did you mean the center of mass?
The center of momentum frame is defined as the frame where the total momentum of the system is 0. The system can have a moving part as long as it also has another part moving with equal and opposite momentum.

The center of momentum frame is defined as the frame where the total momentum of the system is 0. The system can have a moving part as long as it also has another part moving with equal and opposite momentum.
So it occurs at a value where the momentum all adds to 0. How would that work for the clock example?

Say the rest of the solid clock is at rest, and the minute hand is moving, in particular situated at the 12oclock position. So at that position it would have a momentum to the right. Then we would have to say that the center of mass moves to the left so that the total momentum cancels. Is that correct?

Then would the frame keep changing since we have to look at frames where the momentum of the cm of the clock cancels out with the tangential momentum of the clock?

Also, this analysis would not work for a clock placed on a table on earth? Since that clock would be accelerating through spacetime due to the repulsive pressure of the table.

The center of mass is a point. The object has 0 momentum, not the center of mass.
The total momentum doesn't have to be 0. In which case, you can calculate the invariant mass using
$m = \sqrt{\frac{E^2}{c^4} - \frac{p^2}{c^2}}$
By going into the frame where momentum is 0, it's simpler: $m = \frac{E}{c^2}$
But you get the same result regardless of frame. (That's why it's called "invariant")
So you are saying that the center of mass frame is not the same as the center of momentum frame.

Can I use the center of mass frame if there are no external forces on the system? Because in that frame, the momentum of the system should be 0, classically speaking.

Khashishi
Center of mass frame and center of momentum frame mean the same thing. In either case, the center of mass doesn't move.

Say the rest of the solid clock is at rest, and the minute hand is moving, in particular situated at the 12oclock position. So at that position it would have a momentum to the right. Then we would have to say that the center of mass moves to the left so that the total momentum cancels. Is that correct?
No. In the center of momentum frame, if the minute hand is moving right, that means the rest of the clock must be moving left, such that the total momentum is 0.

Center of mass frame and center of momentum frame mean the same thing. In either case, the center of mass doesn't move.

No. In the center of momentum frame, if the minute hand is moving right, that means the rest of the clock must be moving left, such that the total momentum is 0.
Oh, right. I have to think the clock and the hand as two different objects pushing off of each other, where the center of mass of the clock hand system doesn't move. I should say the center of mass of the rest of the clock would move to the left.

Khashishi
Can I use the center of mass frame if there are no external forces on the system? Because in that frame, the momentum of the system should be 0, classically speaking.
Actually, that's the only time you can use the center of mass frame. If there are any external forces, the center of mass frame wouldn't be an inertial reference frame, and we wouldn't use the concept at all.

Actually, that's the only time you can use the center of mass frame. If there are any external forces, the center of mass frame wouldn't be an inertial reference frame, and we wouldn't use the concept at all.
Right, because that would fall under general relativity.

Dale
Mentor
Say the rest of the solid clock is at rest, and the minute hand is moving, in particular situated at the 12oclock position. So at that position it would have a momentum to the right.
So that frame would not be the center of momentum frame. The rest of the clock has a momentum of 0 and the minute hand has nonzero momentum to the right, so the total momentum of the system is to the right.

Also, this analysis would not work for a clock placed on a table on earth? Since that clock would be accelerating through spacetime due to the repulsive pressure of the table
Well, I wouldn't call it an "analysis", it is just a reference frame. You can always define the center of momentum frame for any system. If there is a net external force on the system then that frame is non inertial.