Special relativity change in rest mass

[deadscientist]

A nucleus of rest mass m initially at rest in the lab absorbs a photon such that its total energy becomes 1.01mc^2.

I've calculated the energy of the photon to be greater than the change in rest mass of the nucleus this is on track because the follow up question is why is the required energy of the photon greater than the change in rest mass and I'm not quite sure why this is so please help thank you kindly.

 

[TSny]

Hello.

Does the nucleus stay at rest? Besides energy, what else is conserved?

 

[deadscientist]

Hello.

Does the nucleus stay at rest? Besides energy, what else is conserved?

That is all that is given I assumed it didn't stay at rest because it has a total energy now instead of a rest energy.

 

[deadscientist]

Hello.

Does the nucleus stay at rest? Besides energy, what else is conserved?

The first part asks to find the energy of he photon required to produce this excitation

 

[TSny]

Total energy of the system must be conserved. Try to use this fact to find the energy of the photon using the information given. You should be able to express the energy in terms of the mass m of the nucleus.

 

[deadscientist]

Total energy of the system must be conserved. Try to use this fact to find the energy of the photon using the information given. You should be able to express the energy in terms of the mass m of the nucleus.

I have done this already thanks though my question is why is the energy of the photon greater than the change in rest mass of the nucleus

 

[TSny]

Ok, sorry. So, you have already calculated the energy of the photon and the change in rest mass of the nucleus.

How did you calculate the change in rest mass?

It seems to me that if you understand how to calculate the change in rest mass, then you should be able to see why the photon energy is greater than the change in rest mass energy of the nucleus.

 

[deadscientist]

Ok, sorry. So, you have already calculated the energy of the photon and the change in rest mass of the nucleus.

How did you calculate the change in rest mass?

It seems to me that if you understand how to calculate the change in rest mass, then you should be able to see why the photon energy is greater than the change in rest mass energy of the nucleus.

Perhaps I did it in incorrectly but I used the invariance of the rest mass; E^2=(pc)^2 + (mc^2)^2 knowing E and mc^2 solved for pc which is energy of photon?

 

[deadscientist]

Ok, sorry. So, you have already calculated the energy of the photon and the change in rest mass of the nucleus.

How did you calculate the change in rest mass?

It seems to me that if you understand how to calculate the change in rest mass, then you should be able to see why the photon energy is greater than the change in rest mass energy of the nucleus.

I got an energy of 0.14 times the rest mass

 

[TSny]

I got an energy of 0.14 times the rest mass

This is the photon energy or the change in rest mass energy? Either way, I don't think that is correct. Can you please show your work?

 

[TSny]

Perhaps I did it in incorrectly but I used the invariance of the rest mass; E^2=(pc)^2 + (mc^2)^2 knowing E and mc^2 solved for pc which is energy of photon?

I would have to see what numbers you are substituting into this equation. Please show the details.

But, there is a much easier way to get the photon energy. How would you write the total energy of the system before the photon is absorbed? What is the total energy of the system after the photon is absorbed?

 

[deadscientist]

This is the photon energy or the change in rest mass energy? Either way, I don't think that is correct. Can you please show your work?

Sure

(Cp)^2= (1.01mc^2)^2 - (mc^2)^2
= 1.0201m^2c^4 - m^2c^4
(cp)^2= m^2c^4(1.0201-1)
(cp)^2= 0.0201m^2c^4
cp= 0.1418mc^2

 

[TSny]

Note that you put in 1.01 mc² for the total energy. So, that's the energy of the nucleus after the photon is absorbed. But you put in m for the rest mass. m is the rest mass of the nucleus before the photon was absorbed. So, you have an inconsistency here.

 

[deadscientist]

Note that you put in 1.01 mc² for the total energy. So, that's the energy of the nucleus after the photon is absorbed. But you put in m for the rest mass. m is the rest mass of the nucleus before the photon was absorbed. So, you have an inconsistency here.

I'm confused, 1.01 is the gamma value for the total energy, anyway say I did my calculation correctly, why would the photon have more energy than the change in rest mass? Your help so far has been greatly appreciated

 

[TSny]

In order for the change in rest mass energy to equal the photon energy, all of the photon energy would need to be transformed into rest mass energy. But, think about the nucleus after it absorbs the photon. Will the nucleus be at rest or will it "recoil"? It's like you are standing at rest on a frictionless surface when someone throws you a ball.

 

[deadscientist]

In order for the change in rest mass energy to equal the photon energy, all of the photon energy would need to be transformed into rest mass energy. But, think about the nucleus after it absorbs the photon. Will the nucleus be at rest or will it "recoil"? It's like you are standing at rest on a frictionless surface when someone throws you a ball.

I agree that's why I figured the nucleus will have some kinetic energy and momentum after the photon is absorbed, does it have anything to do with the photon being massless so it requires a much greater energy to change to momentum of a nucleus that has mass

 

[jtbell]

Note that you put in 1.01 mc² for the total energy. So, that's the energy of the nucleus after the photon is absorbed. But you put in m for the rest mass. m is the rest mass of the nucleus before the photon was absorbed. So, you have an inconsistency here.

It may be that the problem was actually stated that way! deadscientist, can you confirm whether the problem statement says the final total energy of the nucleus (which I'll call E') is supposed to be:

E' = 1.01mc^2 (using the initlal rest mass)

or

E' = 1.01m'c^2 (using the final rest mass)

I would ask the instructor/professor about this if necessary, just to make sure.

 

[deadscientist]

It may be that the problem was actually stated that way! deadscientist, can you confirm whether the problem statement says the final total energy of the nucleus (which I'll call E') is supposed to be:

E' = 1.01mc^2 (using the initlal rest mass)

or

E' = 1.01m'c^2 (using the final rest mass)

I would ask the instructor/professor about this if necessary, just to make sure.

Yes you are correct sorry I did not specify the total energy is 1.01 times the initial rest mass (your first suggestion)

 

[TSny]

I agree that's why I figured the nucleus will have some kinetic energy and momentum after the photon is absorbed, does it have anything to do with the photon being massless so it requires a much greater energy to change to momentum of a nucleus that has mass

If you realize that the nucleus will have some kinetic energy after absorbing the photon, then that's the answer to why the increase in rest mass energy is less than the photon energy. Some of the photon energy increases the rest mass energy and some of the photon energy is used to give the nucleus some kinetic energy. So not all of the photon energy is transformed into rest mass energy.

But I think you will need to rethink your calculation. Unless I'm misinterpreting the question, you should be able to see what the energy of the photon is without doing any calculation.

 

[deadscientist]

If you realize that the nucleus will have some kinetic energy after absorbing the photon, then that's the answer to why the increase in rest mass energy is less than the photon energy. Some of the photon energy increases the rest mass energy and some of the photon energy is used to give the nucleus some kinetic energy. So not all of the photon energy is transformed into rest mass energy.

But I think you will need to rethink your calculation. Unless I'm misinterpreting the question, you should be able to see what the energy of the photon is without doing any calculation.

I think you are suggesting that the energy of the photon should be 0.01 times the rest mass based off conservation of energy or am I wrong?

 

[jtbell]

I think you are suggesting that the energy of the photon should be 0.01 times the rest mass based off conservation of energy

Bingo! (now that we've clarified that E' is indeed supposed to be 1.01mc^2, using the initial rest mass)

It's actually possible to get another (different) solution, assuming that E' = 1.01m'c^2, using the final rest mass. It would make a nice "extra credit" problem.

 

[deadscientist]

Bingo! (now that we've clarified that E' is indeed supposed to be 1.01mc^2, using the initial rest mass)

It's actually possible to get another (different) solution, assuming that E' = 1.01m'c^2, using the final rest mass. It would make a nice "extra credit" problem.

Now I'm really confused cause the follow up question asks why the photon energy is more than the change in rest mass (0.01mc^2) so maybe the question is the "extra credit" problem?!?

 

[TSny]

I think you are suggesting that the energy of the photon should be 0.01 times the rest mass based off conservation of energy or am I wrong?

Yes, the photon energy is .01mc². Doesn't that follow from conservation of energy?

 

[deadscientist]

Yes, the photon energy is .01mc². Doesn't that follow from conservation of energy?

I agree but the follow up says the energy of the photon is greater than the change in rest mass can you clarify this

 

[TSny]

After the photon is absorbed, the nucleus has increased its rest mass energy as well as gained kinetic energy. Where did the gain in rest mass energy come from? Where did the gain in KE come from?

 

[jtbell]

Now I'm really confused cause the follow up question asks why the photon energy is more than the change in rest mass (0.01mc^2) so maybe the question is the "extra credit" problem?!?

No. You're given that the final (total) energy of the nucleus is E' = 1.01mc^2, where you use the initial mass. This is just for the purpose of giving you a value for it. That energy is also given by E' = γm'c^2 where m' is the final mass, and γ is the Lorentz factor that comes from the final speed of the nucleus. γ does not equal 1.01, and m' does not equal m.

E' = γm'c^2 is not very useful for you in this case, however. What's more useful is the general relationship between energy, mass and momentum for the final nucleus, which is... <expectant pause>

(This is assuming you want to find the final mass of the nucleus, so you can find the difference from the initial mass. But if all you need to do is explain why the change in rest-mass energy has to be less than the photon energy, without actually calculating it, then consider TSny's hint.)

 

[deadscientist]

After the photon is absorbed, the nucleus has increased its rest mass energy as well as gained kinetic energy. Where did the gain in rest mass energy come from? Where did the gain in KE come from?

After thinking about it more and more I'm just having more and more question I appreciate your patience and help so far, I was taught that rest mass is invariant which makes me question why the nucleus even has a change in 'rest mass' unless this rest mass taken to be from the frame of reference of the nucleus' rest frame where he appears to be still and the world moves towards it. If it does gain rest mass this 'rest mass' would actually be its original rest mass plus the kinetic energy it has acquired because it now has momentum?! if this is so one could see the relativistic mometum of a particle and its kinetic energy can be thought of as both contributing to the change in its rest mass? The acquired kinetic energy comes from conservation of energy the initial energy is the rest energy of the nucleus at rest in both its frame and the lab fram plus the energy of the photon, the final energy is equal to the total energy given. Using this gives a photon energy of .01 times the original rest mass times the speed of light squared. The nucleus change in rest mass is less than this because the change in its proper rest frames mass is contributed to by both its momentum AND kinetic energy??

 

[TSny]

Rest mass is the same as invariant mass. “Invariant” means that rest mass has the same value in all inertial reference frames. It does not mean that the rest mass cannot change.

For example, a nucleus can absorb energy from a photon and some of that energy can be stored “internally” as excitation energy of the nucleons. The energy of the excited nucleus when it is at rest is greater than the energy of the unexcited nucleus when it is at rest. This increase in “rest” energy shows up as an increase in rest mass (or invariant mass) of the nucleus (from m to m’). All observers will agree that the new invariant mass is m’.

In a reference frame in which the excited nucleus is moving, the nucleus will have a total energy E’ given by the rest mass energy m’c² plus the kinetic energy due to motion: E’ = m’c² + KE'. Or, in terms of momentum: E’² = (p’c)²+(m’c²)² . The primes indicate “after the photon is absorbed”.

In your problem, the nucleus absorbs energy from the photon and recoils with momentum p’, total energy E’, and rest mass m’.

Conservation of energy says that the total energy E of the system before the photon is absorbed must equal the total energy E’ after the photon is absorbed.

Conservation of momentum says that the total momentum p of the system before equals the total momentum p’ after.

 

[jtbell]

Rest mass is the same as invariant mass. “Invariant” means that rest mass has the same value in all inertial reference frames. It does not mean that the rest mass cannot change.

For example, a nucleus can absorb energy from a photon and some of that energy can be stored “internally” as excitation energy of the nucleons. The energy of the excited nucleus when it is at rest is greater than the energy of the unexcited nucleus when it is at rest. This increase in “rest” energy shows up as an increase in rest mass (or invariant mass) of the nucleus (from m to m’). All observers will agree that the new invariant mass is m’.

For a macroscopic example, consider a brick which has a certain rest mass (invariant mass). Supply heat to it and raise its temperature by increasing its thermal energy. Its rest mass increases according to the amount of thermal energy that it gains.

The rest mass remains constant only for a truly fundamental, indivisible particle such as an electron or muon. A system of particles can change its rest mass by absorbing or losing internal energy.

 

[deadscientist]

I think you are suggesting that the energy of the photon should be 0.01 times the rest mass based off conservation of energy or am I wrong?

Thankyou both very much truly very helpful :)