Questions from Halliday

[irockhard]

As we watch, a spaceship passes us in time Δt. The crew of the spaceship measures the passage time and finds it to be Δt'. Which of the following statements is true?
A. Δt is the proper time for the passage and it is smaller than Δt'
B. Δt is the proper time for the passage and it is greater than Δt'
C. Δt' is the proper time for the passage and it is smaller than Δt
D. Δt' is the proper time for the passage and it is greater than Δt
E. None of the above statements are true.


The answer according to the answer key is C. But I am 100% convinced that it's A.
Because proper time is the time in the reference frame in which the two events
occur at the same place. And in this problem, the observer sees the two events
at the same place while the crew sees the two events across the length of the
spaceship.

[blackcat007]

As we watch, a spaceship passes us in time Δt. The crew of the spaceship measures the passage time and finds it to be Δt'. Which of the following statements is true?
A. Δt is the proper time for the passage and it is smaller than Δt'
B. Δt is the proper time for the passage and it is greater than Δt'
C. Δt' is the proper time for the passage and it is smaller than Δt
D. Δt' is the proper time for the passage and it is greater than Δt
E. None of the above statements are true.


The answer according to the answer key is C. But I am 100% convinced that it's A.
Because proper time is the time in the reference frame in which the two events
occur at the same place. And in this problem, the observer sees the two events
at the same place while the crew sees the two events across the length of the
spaceship.

I find this ambiguous, passage of what the spaceship (of some length)? or our frame (of some length)?
consider the following two cases of passage:
we (say S) will measure the time interval(delta t in which the length of the spaceship traverses us) while standing still, of the two events of the front and end of the spaceship passing our position ,
while for the crew (say S'), they will have to setup synchronized clocks at the front and the end of the spaceship to measure the time interval (delta t') between S passing the front and the end.
where delta t = gamma*(delta t' -u delta x'/c^2)
here S measures the proper time.

BUT!!! consider the following case:
S is standing on a platform of length L, and the passage of our platform is considered as the "passage" , in this case the entire scenario is inverted, S has to setup clocks at the two ends of the platform to measure the passage of the spaceship along the length of the platform while S' can measure the passage of the platform by a single clock, standing at the same location in his frame, in this case S' measures the proper time.

[physics_auth]

As we watch, a spaceship passes us in time Δt. The crew of the spaceship measures the passage time and finds it to be Δt'. Which of the following statements is true?
A. Δt is the proper time for the passage and it is smaller than Δt'
B. Δt is the proper time for the passage and it is greater than Δt'
C. Δt' is the proper time for the passage and it is smaller than Δt
D. Δt' is the proper time for the passage and it is greater than Δt
E. None of the above statements are true.


The answer according to the answer key is C. But I am 100% convinced that it's A.
Because proper time is the time in the reference frame in which the two events
occur at the same place. And in this problem, the observer sees the two events
at the same place while the crew sees the two events across the length of the
spaceship.

I think that more information need to be given so as to decide between A and C. Anyway ... if we assume that the observer on the ground (i.e. at S), let us call him O, focuses on one point in space using his own clockwatch, then it is true that his clockwatch gives the proper time. At the same time the crew (i.e. the observer at S', let us call him O') uses as a reference point observer O on the ground (i.e. at S), but they use two sychronized -for them only- clocks. They use one clock T at the front of the spaceship to measure the time at which observer O passes the front side of their spaceship and another clock T' on the rear of the spaceship to measure the time at which observer O (again) passes the rear of their spaceship, thus they don't measure the proper time -> this is because the two events occur at different points in space for observer O' (and at system S') and so cannot be separated by a proper time in system S'. If this what is implied in the question ... then A is the correct answer.

Physics_auth

[irockhard]

Thanks guys! here I have another question

During an adiabatic process an object does 100 J of work and its temperature decreases by 5K.During another process it does 25 J of work and its temperature decreases by 5K. Its heat capacity for the second process is:

A. 20 J/K
B. 24 J/K
C. 5 J/K
D. 15 J/K
E. 100 J/K

My first instinct is to go with C = dU/dT = -100/-5 = 20 J/k for the 1st process and same for the second process. But my better instinct tells me that this is definitely wrong.

[blackcat007]

Thanks guys! here I have another question

During an adiabatic process an object does 100 J of work and its temperature decreases by 5K.During another process it does 25 J of work and its temperature decreases by 5K. Its heat capacity for the second process is:

A. 20 J/K
B. 24 J/K
C. 5 J/K
D. 15 J/K
E. 100 J/K

My first instinct is to go with C = dU/dT = -100/-5 = 20 J/k for the 1st process and same for the second process. But my better instinct tells me that this is definitely wrong.

Is it D?

for the first case dQ=dW+dU dQ=0 =>nCv(-5)=-100 => nCv=20 (but this is not the heat capacity)
heat capacity is delta(Q)/delta(T)=C

thus for the second process dQ=C*(-5)=25+(20)(-5) => C=75/5=15J/K
heat capacity is the heat stored by a given mass of substance, thus unlike specific heat it is not constant, since heat capacity is sp heat*mass/moles (depending on the units of sp heat)

[physics_auth]

Thanks guys! here I have another question

During an adiabatic process an object does 100 J of work and its temperature decreases by 5K.During another process it does 25 J of work and its temperature decreases by 5K. Its heat capacity for the second process is:

A. 20 J/K
B. 24 J/K
C. 5 J/K
D. 15 J/K
E. 100 J/K

My first instinct is to go with C = dU/dT = -100/-5 = 20 J/k for the 1st process and same for the second process. But my better instinct tells me that this is definitely wrong.

For an ideal gas U = U(T) and c_v and n = moles have particular values (they do not change). For the same temperature change (i.e. of 5 K) the change in U -> ΔU is the same.
Using the 1st law of thermo for each process we have:
ΔQ = ΔU + ΔW (1) for the adiabatic process
ΔQ' = ΔU' + ΔW' (2) for the other process
but ΔU = ΔU' according to what was said ... and ΔQ = 0 for the adiabatic process ..., thus by (1) and (2) it is
ΔQ' = ΔW' - ΔW (3)
Α heat capacity is generally defined as C = ΔQ/ΔΤ = (25 - 100)J / (-5 K) = 15 J/K. Ι used "-" for ΔΤ since we have a decrease. Thus (D).

! C = ΔU/ΔΤ is not the general definition of C, it holds in isochoric process only, where ΔW = 0. The general definition is C = ΔQ/ΔΤ.

Physics_auth

[irockhard]

Thanks guys! here I have another question

During an adiabatic process an object does 100 J of work and its temperature decreases by 5K.During another process it does 25 J of work and its temperature decreases by 5K. Its heat capacity for the second process is:

A. 20 J/K
B. 24 J/K
C. 5 J/K
D. 15 J/K
E. 100 J/K

My first instinct is to go with C = dU/dT = -100/-5 = 20 J/k for the 1st process and same for the second process. But my better instinct tells me that this is definitely wrong.

Is it D?

for the first case dQ=dW+dU dQ=0 =>nCv(-5)=-100 => nCv=20 (but this is not the heat capacity)
heat capacity is delta(Q)/delta(T)=C

thus for the second process dQ=C*(-5)=25+(20)(-5) => C=75/5=15J/K
heat capacity is the heat stored by a given mass of substance, thus unlike specific heat it is not constant, since heat capacity is sp heat*mass/moles (depending on the units of sp heat)

Can you explain what nCv in your first line means?
Thanks for your help!

[physics_auth]

Thanks guys! here I have another question

During an adiabatic process an object does 100 J of work and its temperature decreases by 5K.During another process it does 25 J of work and its temperature decreases by 5K. Its heat capacity for the second process is:

A. 20 J/K
B. 24 J/K
C. 5 J/K
D. 15 J/K
E. 100 J/K

My first instinct is to go with C = dU/dT = -100/-5 = 20 J/k for the 1st process and same for the second process. But my better instinct tells me that this is definitely wrong.

Is it D?

for the first case dQ=dW+dU dQ=0 =>nCv(-5)=-100 => nCv=20 (but this is not the heat capacity)
heat capacity is delta(Q)/delta(T)=C

thus for the second process dQ=C*(-5)=25+(20)(-5) => C=75/5=15J/K
heat capacity is the heat stored by a given mass of substance, thus unlike specific heat it is not constant, since heat capacity is sp heat*mass/moles (depending on the units of sp heat)

Can you explain what nCv in your first line means?
Thanks for your help!

n = moles
Cv = molar heat capacity at constant volume
see the dimensions on each side

[irockhard]

Thanks guys! here I have another question

During an adiabatic process an object does 100 J of work and its temperature decreases by 5K.During another process it does 25 J of work and its temperature decreases by 5K. Its heat capacity for the second process is:

A. 20 J/K
B. 24 J/K
C. 5 J/K
D. 15 J/K
E. 100 J/K

My first instinct is to go with C = dU/dT = -100/-5 = 20 J/k for the 1st process and same for the second process. But my better instinct tells me that this is definitely wrong.

For an ideal gas U = U(T) and c_v and n = moles have particular values (they do not change). For the same temperature change (i.e. of 5 K) the change in U -> ΔU is the same.
Using the 1st law of thermo for each process we have:
ΔQ = ΔU + ΔW (1) for the adiabatic process
ΔQ' = ΔU' + ΔW' (2) for the other process
but ΔU = ΔU' according to what was said ... and ΔQ = 0 for the adiabatic process ..., thus by (1) and (2) it is
ΔQ' = ΔW' - ΔW (3)
Α heat capacity is generally defined as C = ΔQ/ΔΤ = (25 - 100)J / (-5 K) = 15 J/K. Ι used "-" for ΔΤ since we have a decrease. Thus (D).

! C = ΔU/ΔΤ is not the general definition of C, it holds in isochoric process only, where ΔW = 0. The general definition is C = ΔQ/ΔΤ.

Physics_auth

I assume you didn't have Kittel for thermal physics? I have not realized how much that book sucks or how little I learned from that book until now. To be honest, this is my first time seeing the word "isochoric" and "molar heat capacity".

So my follow up question is how do you justify that "For the same temperature change (i.e. of 5 K) the change in U -> ΔU is the same. " Do you reason that for an idea gas
U = (3/2)nRkT. Also I understood the rest of your solution. But I don't see how n or Cv has anything to do with this question? Am I still missing something?

[physics_auth]

Thanks guys! here I have another question

During an adiabatic process an object does 100 J of work and its temperature decreases by 5K.During another process it does 25 J of work and its temperature decreases by 5K. Its heat capacity for the second process is:

A. 20 J/K
B. 24 J/K
C. 5 J/K
D. 15 J/K
E. 100 J/K

My first instinct is to go with C = dU/dT = -100/-5 = 20 J/k for the 1st process and same for the second process. But my better instinct tells me that this is definitely wrong.

For an ideal gas U = U(T) and c_v and n = moles have particular values (they do not change). For the same temperature change (i.e. of 5 K) the change in U -> ΔU is the same.
Using the 1st law of thermo for each process we have:
ΔQ = ΔU + ΔW (1) for the adiabatic process
ΔQ' = ΔU' + ΔW' (2) for the other process
but ΔU = ΔU' according to what was said ... and ΔQ = 0 for the adiabatic process ..., thus by (1) and (2) it is
ΔQ' = ΔW' - ΔW (3)
Α heat capacity is generally defined as C = ΔQ/ΔΤ = (25 - 100)J / (-5 K) = 15 J/K. Ι used "-" for ΔΤ since we have a decrease. Thus (D).

! C = ΔU/ΔΤ is not the general definition of C, it holds in isochoric process only, where ΔW = 0. The general definition is C = ΔQ/ΔΤ.

Physics_auth

I assume you didn't have Kittel for thermal physics? I have not realized how much that book sucks or how little I learned from that book until now. To be honest, this is my first time seeing the word "isochoric" and "molar heat capacity".

So my follow up question is how do you justify that "For the same temperature change (i.e. of 5 K) the change in U -> ΔU is the same. " Do you reason that for an idea gas
U = (3/2)nRkT. Also I understood the rest of your solution. But I don't see how n or Cv has anything to do with this question? Am I still missing something?

As I remember ... it is U = U(T) = (3/2) nRT (without k). But for the ideal gas (in which by definition the gas particles have only translational degrees of freedom) it can be found that c_v = (3/2)R thus U = U(T) = n*c_v*T. After PGRE test you can reread thermodynamics from Zemanski Dittman ... it is a very enlightening book. As for Kittel I am not a devotee of his works ... I don't like the way he writes.

[prajor]

I think that more information need to be given so as to decide between A and C. Anyway ... if we assume that the observer on the ground (i.e. at S), let us call him O, focuses on one point in space using his own clockwatch, then it is true that his clockwatch gives the proper time. At the same time the crew (i.e. the observer at S', let us call him O') uses as a reference point observer O on the ground (i.e. at S), but they use two sychronized -for them only- clocks. They use one clock T at the front of the spaceship to measure the time at which observer O passes the front side of their spaceship and another clock T' on the rear of the spaceship to measure the time at which observer O (again) passes the rear of their spaceship, thus they don't measure the proper time -> this is because the two events occur at different points in space for observer O' (and at system S') and so cannot be separated by a proper time in system S'. If this what is implied in the question ... then A is the correct answer.

Physics_auth

Sorry, back to the relativity question:

1. Isnt it correct to say since S uses a single clock, his time would be proper ?
2. As far as what is smaller - Since S and S' are relative (you can take anyone as earth and the other as spaceship ), isn't it true that S moves @ speed -v w.r.t S. How do we decide which is the time dilated frame S or S'.
Why this is not symmetric ?