Relativistic Density - Check My Work & Learn Methods

[LilRubyKinz]

Homework Statement

Consider a cube to have a density of 2.0kg/m^3. What is its relativistic density at 0.95c?

Relevant Equations

Lm = Ls/(1-v^2/c^2)^ - 1/2
mm = ms/(1-v^2/c^2)^ - 1/2
Density = mass/volume

We never learned how to use these formulas, so I'm pretty much grasping for straws. I got 2.04 kg/m^3. Can anyone check my work/teach me proper methods if I'm wrong? Thanks.

 

[LilRubyKinz]

Wait that's definitely wrong... Now I'm getting 42.3. Is that right?

 

[PeroK]

Wait that's definitely wrong... Now I'm getting 42.3. Is that right?

Can you explain what you're doing?

 

[LilRubyKinz]

Yes. I'll attach a picture.

 

[LilRubyKinz]

I wrote the last division statement wrong (switch numerator and denominator) but yes I get 42.3 or 45 if I don't round.

 

[jbriggs444]

Homework Statement: Consider a cube to have a density of 2.0kg/m^3. What is its relativistic density at 0.95c?
Homework Equations: Lm = Ls/(1-v^2/c^2)^ - 1/2
mm = ms/(1-v^2/c^2)^ - 1/2
Density = mass/volume

Typesetting those equations for you. See this link to see how it was done.

$$L_m = \frac{L_s}{\sqrt{1-v^2/c^2}}$$
$$m_m = \frac{m_s}{\sqrt{1-v^2/c^2}}$$

One assumes that $L_m$ is the length of the moving cube and $L_s$ is its length as measured from a frame where it is stationary. Similarly, $m_m$ would be the relativistic mass of the moving cube and $m_s$ is its mass as measured from a frame where it is stationary.

Did you forget to square something?

 

[PeroK]

If I can understand what you're doing, you are length contracting all three dimensions of of the cube.

 

[LilRubyKinz]

If I can understand what you're doing, you are length contracting all three dimensions of of the cube.

Yes I think that's what I'm doing. Am I not supposed to? Is this the wrong formula?

 

[jbriggs444]

Length contraction is only in the direction of relative motion.

 

[LilRubyKinz]

So what formula should I be using to get the new volume?

 

[jbriggs444]

So what formula should I be using to get the new volume?

Same one they taught you in grade school. Volume = length * width * height.

 

[LilRubyKinz]

Okay, so I don't need to change anything about length? Just mass changes?

 

[jbriggs444]

Okay, so I don't need to change anything about length? Just mass changes?

What makes you think that length contraction does not apply?

 

[LilRubyKinz]

Length contraction is only in the direction of relative motion.

I confuse easily, sorry. I thought this meant not to use it.

Can you please walk me through the correct process of calculating the new mass and lengths?

 

[jbriggs444]

I confuse easily, sorry. I thought this meant not to use it.

Can you please walk me through the correct process of calculating the new mass and lengths?

You already quoted the correct formula for length contraction. By what factor has the length of the cube contracted?

This factor is also known as gamma ($\gamma$).

 

[LilRubyKinz]

Okay here's the problem. The square root of 1 - (0.95c)^2 / c^2 should be 0.312. I calculated this wrong.

I'm going to try again with this new value; I'll let you know what I get

 

[PeroK]

Okay here's the problem. The square root of 1 - (0.95c)^2 / c^2 should be 0.312. I calculated this wrong.

I'm going to try again with this new value; I'll let you know what I get

Put your calculator away and start thinking instead!

 

[LilRubyKinz]

Is only one side length affected by length contraction or all three?

 

[PeroK]

Is only one side length affected by length contraction or all three?

Who says the cube is moving parallel to one of its edges? But, let's assume it is, what do you think?

 

[LilRubyKinz]

Yes?

 

[PeroK]

Yes?

There is no length contraction in directions perpendicular to the motion.

 

[LilRubyKinz]

There is no length contraction in directions perpendicular to the motion.

So does that mean I'm right? Sorry I'm not great at this!

 

[jbriggs444]

So does that mean I'm right? Sorry I'm not great at this!

Right about what?

 

[PeroK]

So does that mean I'm right? Sorry I'm not great at this!

Yes, only one side is contracted.

 

[LilRubyKinz]

Yes, only one side is contracted.

Okay thank you!

Now one more thing I don't understand.

mm = 2 / 0.312 = 6.4 m
Lm = 1 / 0.312 = 3.2 kg

6.4 / 3.2(1)(1) = 2

Why am I still getting 2 as the answer? Does the density not change?

 

[jbriggs444]

Okay thank you!

Now one more thing I don't understand.

mm = 2 / 0.312 = 6.4 m
Lm = 1 / 0.312 = 3.2 kg

6.4 / 3.2(1)(1) = 2

Why am I still getting 2 as the answer? Does the density not change?

Can you compare your formula for length contraction to the one in the book? When length "contracts", does it get larger or smaller?

I am also concerned about your units. Your contracted length is expressed in kilograms?

 

[LilRubyKinz]

So I am supposed to multiply there, not divide? Making the new length 0.312?

Making the answer 20.5?

Or is density supposed to be unchanged??

My brain hurts.

 

[LilRubyKinz]

Can you compare your formula for length contraction to the one in the book? When length "contracts", does it get larger or smaller?

I am also concerned about your units. Your contracted length is expressed in kilograms?

Sorry, swap kg and m there. Copied it wrong.

But the book shows Lm and Ls reversed from what I have here...

So 20.5 should be the answer? Or 2?

 

[jbriggs444]

Go back to basics.

Moving objects appear to be shortened. Short objects have lower volume. Lower volume means higher density.

Moving objects gain relativistic mass. [Note that relativistic mass is a deprecated concept that would better be left untaught]. More mass means higher density.

Both effects call for an increased density. So an answer of 2.0 kg/m³ could not be correct.

20.5 kg/m³ matches what I get. Though you might want to consider rounding to an appropriate number of significant figures.

 

[LilRubyKinz]

Both effects call for an increased density. So an answer of 2.0 kg/m³ could not be correct.

Okay, getting a little bit of mixed messages here... But I think 20.5 makes the most sense to me. Thank you very much! I will fix significant digits now.

 

[jbriggs444]

I think @PeroK and myself are on the same page here. If you perceive a conflict, either of us can probably clarify it away.

 

[LilRubyKinz]

Okay thank you!

 

[PeroK]

I think @PeroK and myself are on the same page here. If you perceive a conflict, either of us can probably clarify it away.

Yes, I agree.

I was distracted trying to find a way to show that it's true in general, even when motion is not in the same direction as one side of the cube. Which I've just spotted!

 

[jbriggs444]

I was distracted trying to find a clever way to show that it's true in general, even when motion is not in the same direction as one side of the cube. Which I've just spotted!

I was taking it as an obvious geometric property -- scale down one dimension by a factor of $\gamma$ and the volume clearly goes down proportionately.

I gave a moment's though to Terrell rotation, but it does not apply.

 

[PeroK]

I was taking it as an obvious geometric property -- scale down one dimension by a factor of $\gamma$ and the volume clearly goes down proportionately.

Yes, I was only thinking about cubes, parallelopipeds and the triple scalar product! Then, I realized ...