(E,p)(E,-p) What pushes on a scale?

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In summary, particles are typically weighed in their rest frame, where the momentum is measured as the first component of the momentum vector. However, in the case of non-zero momentum, the particle will not stay in one's hand for long. Under restricted conditions, the particle's mass can be defined as its gravitational mass. The 3-force is what the experimenter applies to the object, and the scale measures the force with which it pushes. The particle does not push on the scale, but rather the scale pushes on the particle. The equations for force and acceleration in special relativity can be derived from the 4-momentum vector.
  • #1
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You hold in the palm of your hand a classical particle with momentum

[tex]P = (E,cp_{x},cp_{y}, cp_{up})[/tex] where [tex]cp_{up}=0[/tex] .

Placing it gently on a scale you ask, what part of

[tex]P = (E,cp_{x},cp_{y}, cp_{up})[/tex]

is pushing down on the scale?
 
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  • #2
Usually you weigh particles in their rest frame, otherwise they move around too much =) In that case, you are measuring the first component of [tex]P[/tex].

You might consider the following: A gas of such non-interacting particles, in a box. The average 4-momentum of the fluid is then [tex]\left\langle P \right\rangle = ( \left\langle E \right\rangle, 0, 0, 0)[/tex], so the mass if the gas is going to be [tex]\left\langle E\right\rangle[/tex].
 
  • #3
What if the momentum is non-zero?
 
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  • #4
Then it won't be in the palm of your hand for very long.
 
  • #5
OK, so backing up a little, in everyday life, where things sit still on a scale.

[tex]\langle P \rangle ^2 = m^2 = E^2/c^4 - p^2/c^2[/tex], defining m after some units adjustment, and obtaining your equation when [tex]p=0[/tex] .

Under restricted conditions, m is what could be called gravitational mass.

But I think you misunderstand me; I really don't know what the general solution is.

I can guess halfway through it. Using the equivalence principle, the problem is equivalent to asking what resists acceleration a_z, by a force f_z, in the lab frame. So I might guess that it involves the 4-vectors of force and acceleration, whatever they are. Do you think so?
 
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  • #6
I have some notes which might help:

http://www.physics.thetangentbundle.net/wiki/Special_relativity/force [Broken]
http://www.physics.thetangentbundle.net/wiki/Special_relativity/motion_under_constant_force [Broken]

The "3-force" is what the experimenter applies to the object. The measurement on the scale, if you will.
 
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  • #7
I would say that neither E nor p push on the scale. Instead, the scale pushes on the particle and measures the force with which it pushes.
 
  • #8
You've got a point Dale. If the scale acts with Fz in the lab frame, what dv/dt, in either frame, does the particle have?
 
  • #9
A quick perusal of the links I gave would show that the answer to that question is
[tex]\vec{f} = \frac{1}{\gamma} \frac{d^2 \vec{x} }{d\tau^2} = \frac{1}{\gamma} \frac{d}{d\tau} \left( \gamma \frac{d\vec{x}}{dt} \right) = \frac{d}{dt} \left( \gamma \frac{d\vec{x}}{dt} \right)[/tex]

Cmon people... no need for spoon feeding.
 
  • #10
lol, lbrits. I need spoon feeding. To be honest, the equations in those links were pretty haphazardly done, but for review purposes. The deductive chain was obscure.
 
  • #11
The first link assumes basic familiarity with SR, but I've tried to make it more self contained. Thanks.
 
  • #12
Sorry, I didn't know you wrote it. I hope you can take it as constructive criticism from someone who hasn't delt with SR in a while, and didn't devote a great deal of time reading it. To be fair, I've been involved in another persuit that takes up what little brain power my wee brain can handle.
 
  • #13
Phrak said:
You've got a point Dale. If the scale acts with Fz in the lab frame, what dv/dt, in either frame, does the particle have?
Since you are already starting with the http://en.wikipedia.org/wiki/Four-momentum" [Broken] is simply dp/dtau.
 
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1. What is (E,p)(E,-p)?

(E,p)(E,-p) is a mathematical representation of the conservation of momentum, where E represents energy and p represents momentum. It shows that in a closed system, the total momentum before and after an interaction remains the same.

2. How does (E,p)(E,-p) relate to pushing on a scale?

Pushing on a scale involves a transfer of momentum from an object to the scale. According to (E,p)(E,-p), the total momentum before and after the push must be the same. This means that the scale will register a change in momentum, and therefore a change in weight, when an object is placed on it.

3. What is the significance of (E,p)(E,-p) in physics?

(E,p)(E,-p) is a fundamental principle in physics, as it demonstrates the conservation of momentum in all interactions. It is used to explain and predict the behavior of objects in motion and is an important concept in fields such as mechanics, thermodynamics, and electromagnetism.

4. How does (E,p)(E,-p) apply to real-life situations?

(E,p)(E,-p) applies to all interactions between objects, whether it is a collision between two billiard balls or a person pushing a shopping cart. It helps to explain the forces involved and the resulting motion of the objects involved.

5. Are there any exceptions to (E,p)(E,-p)?

While (E,p)(E,-p) holds true in most scenarios, there are some situations where it may not apply. This includes interactions involving electromagnetic fields or objects traveling at speeds close to the speed of light. In these cases, the principles of relativity must be taken into account.

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