Exploring the Concept of Effective Inertia in Special Relativity

[rpthomps]

Hello,

I have the Perimeter Institute Activity Package on Special Relativity and I am working through the problems to teach to my students and I am stuck on the following scenario:

Alice open a box and finds two balls of mass m connected by a spring. The balls are oscillating back and forth very quickly. When they are moving fastest, their time dilation factor is γ. Using the concept of "effective inertia" for sideways deflection, how does the motion of the balls affect the mass of the box? Does kinetic energy have inertia?

The answer for this is given and reads:

The moving balls have more effective inertia for sideways deflection, which means the box will present a greater resistance to upward acceleration.(It can be shown that this effect is the same for all directions of acceleration) The kinetic energy of the balls increases the inertia (or mass) of the box.

Now, the point of this question is to get students to think of mass as energy and that an increase in velocity translates into an increase in mass. However, I am not confidant that students with a very new introduction to time dilation would arrive at this conclusion without significant prompting. Is there a way of rephrasing the above question that draws on more conventional ideas and experiences of a typical physics student? Or do you think it is necessary to talk "teach" how momentum changes as velocity does before getting them to think about this question?

Always appreciated,

Ryan

 

[Simon Bridge]

It is with some dismay that I read that since getting away from the idea of "relativistic mass" is something of a struggle later. I've been trying to find another reference for you - but no luck.
I think they are trying to simulate a binding energy effect using the spring.

Now, the point of this question is to get students to think of mass as energy and that an increase in velocity translates into an increase in mass. However, I am not confidant that students with a very new introduction to time dilation would arrive at this conclusion without significant prompting.

They won't - it is usually produced well after the basic Lorentz transformations.

Is there a way of rephrasing the above question that draws on more conventional ideas and experiences of a typical physics student?

No. In fact that is a bad idea - we need relativity because those "more conventional ideas" are not good enough.

Or do you think it is necessary to talk "teach" how momentum changes as velocity does before getting them to think about this question?

Yes.

... how would you normally measure the box's inertia?
Use the new knowledge of relativity to crunch the numbers.

But if the goal is just to get students thinking that mass is a form of energy - the usual path is through the energy-momentum relationship.

You can discover that energy is related to mass via a Universal constant ... which you tell them means that energy and mass are aspects of the same stuff.

$E_{tot}=\gamma mc^2$ and $E_K=(\gamma -1)mc^2$
... as $v\rightarrow 0$ $E_{tot}\rightarrow mc^2$
... m is the "invariant" or "rest" mass.

also see:
http://www.aip.org/history/einstein/voice1.htm
http://plato.stanford.edu/entries/equivME/

 

[bcrowell]

It would be helpful to have a link for the materials you're referring to and a description of what level of students we're talking about (high school?).

Just for background, the way I would look at this is in terms of the energy-momentum four-vector. If the two balls are moving in opposite directions, then their energy-momentum four-vectors look like (E,p) and (E,-p). The magnitude of each of these is $m=\sqrt{E^2-p^2}$. The total energy-momentum four-vector is (2E,0), and its magnitude is 2E, which is greater than 2m.

 

[rpthomps]

As always, your interest and help is much appreciated.

Here is the link to the doc file you requested.

http://www.perimeterinstitute.ca/images/perimeter_inspirations/Revolutions/11_sw2-sci-revolution-special-relativity.doc

And yes, the level of the students is high school.

 

[bcrowell]

Here is the link to the doc file you requested.

http://www.perimeterinstitute.ca/images/perimeter_inspirations/Revolutions/11_sw2-sci-revolution-special-relativity.doc

What is the context? Clearly this is part of some larger project...?

 

[rpthomps]

The activity is designed to be an introduction to the concept of relativity. Rather than explicitly teaching the concept, the teacher puts the students into small groups and has them work together coming up with their own answers. Each question builds on the ideas of the previous question. The challenge I am observing is that some of the concepts seem rather too difficult to allow students to just arrive at the answer by themselves. I would need to intervene and show them the idea first, which makes parts of the activity useless.

 

[rpthomps]

My preference would be to modify the activity and provide questions that students could do on their own and then discuss as a bigger group the ideas inherent in the activity which is relativity. This is the basis for my original question.

 

[Naty1]

The activity is designed to be an introduction to the concept of relativity. Rather than explicitly teaching the concept, the teacher puts the students into small groups and has them work together coming up with their own answers.

I don't know just what that means, but considering all the difficulty some of the finest minds in science had with 'luminiferous ether'...it seems some introduction to SR with some basic math relationships via a teacher is appropriate.

Einstein's own book, Relativity, the special and general theory offers a logical sequence using basic algebra...

...some of the concepts seem rather too difficult to allow students to just arrive at the answer by themselves.

I'll bet...

 

[rpthomps]

I think what I am going to do is go over Feynman's own lecture on relativistic mass using conservation of momentum. When I finish the lesson I will post it here for interest sake. Thank you all for your thoughts.

 

[Simon Bridge]

Those sorts of exercizes are great if you have a lot of time and don't mind the students coming up with a load of rubbish or quickly getting bogged down. You are correct that students will not come up with special relativity by themselves - the exercise can, with guidance, illustrate to them why it is needed though.

Once they have struggled to various effect - mostly fruitless - with the concepts, the way is paved for a more teacher-directed expositional approach.

The student directed approach is very powerful but I don't think the resource you have been trying to use is going to be effective.