Title

[u/L31N0PTR1X]

[u/Digital_001]

Observed by who - the kids or onlookers not on the roundabout?

[u/HunsterMonter → u/Digital_001]

By any reference frame! Each will have its own specific connection coefficients but the trajectory will always be given by the geodesic equation (as long as the ball isn't charged, then you need to add the Lorentz force).

[u/Digital_001 → u/HunsterMonter]

That's a lot of possible reference frames! This is gonna be a long proof...

[u/HunsterMonter → u/Digital_001]

Better start now then 😁 (It requires a bit of general relativity knowledge, but you can find some proofs of the geodesic equation here)

[u/Digital_001 → u/HunsterMonter]

This is a good start, but I feel we should include drag and wind if we are to find the ball's true path - they would have a significant effect is the ball is light. The Magnus effect could feature too.

The ball's acceleration downwards due to gravity (in the ground's frame) is, of course, already included in general relativity.

[u/HunsterMonter → u/Digital_001]

Drag? What is this, r/engineeringmemes?

[u/sneakpeekbot → u/HunsterMonter]

Here's a sneak peek of /r/engineeringmemes using the top posts of the year!

#1: Behold: the inventor of the motorcycle | 130 comments
#2: Two very different units | 41 comments
#3: thoughts? | 63 comments

---

^{I'm a bot, beep boop | Downvote to remove | Contact | Info | Opt-out | GitHub}

[u/kashyou → u/Digital_001]

pick one then use the christoffel symbol transformation law

[u/luisgdh → u/Digital_001]

You can do it for two reference frames and generalize the remaining ones. Just don't tell mathematicians.

[u/L31N0PTR1X → u/HunsterMonter]

Exactly, if this were a question I had asked in a hypothetical exam, I'd accept a student definiting a frame stationary with respect to the roundabout, in Cartesian coordinates, yielding completely trivial connections and just showing that the equation was \ddot{x}^mu =0

Also, if we were in 3+1 dimensions, there's a fun way to add the Lorentz force to the geodesic equation, you can append S¹ to every point in spacetime and then define a connection (that ends up being your electromagnetic gauge field) if you now compute the geodesic, the Lorentz force comes in for free!

[u/FreePeeplup → u/L31N0PTR1X]

No, it’s the frame stationary with respect to the roundabout that will yield non-zero Christoffel symbols, and it is exactly what the question is asking us to find.

The frame that’s stationary with respect to the ground will observe the geodesic equation to be \ddot{x} = 0 with zero Christoffel symbols.

[u/Old-Administration-9 → u/FreePeeplup]

Yep. The ball travels in a straight line with respect to the ground, it only rotates relative to the roundabout!

[u/L31N0PTR1X → u/FreePeeplup]

Oops, jumbled up my words, you know what I meant anyway, after all I wrote the question lmao

Also, that's not what I was asking you to find. I was just asking you to find the expression in a frame, not any particular one. The easy way to the answer here would demonstrate a student's understanding of covariant physics

[u/FreePeeplup → u/HunsterMonter]

The question doesn’t ask to show that the geodesic equation holds for any reference frame in any possible configuration with generic Christoffel coefficients. It asks to find the specific Christoffel coefficients for the specific configuration of a rotating frame and a ball thrown directed towards a diametrically opposed point

[u/what_could_gowrong]

comment

[u/L31N0PTR1X → u/what_could_gowrong]

Touché

[u/SPEC7RE3 → u/what_could_gowrong]

Argumentative reply

[u/suskio4 → u/SPEC7RE3]

Funny image referencing a singular line in the previous message of this thread

[u/Such-Shop-9724]

if you throw a ball while spinning the ball is still going to follow a straight path and the tragectory of a falling abject is gonna follow a parabola (approximatly)

then you could test if the function is a quadratic (which it most likely is) and find gamma

[u/FreePeeplup → u/Such-Shop-9724]

No, it will not appear straight from the co-rotating frame with respect to the roundabout, which is what the question is asking us

[u/L31N0PTR1X → u/FreePeeplup]

To be fair, the question doesn't state which frame you should compute, just that you should compute one. You could just say that you're observing the rotating roundabout in a rectilinear coordinate system and just argue that all connections are trivial, giving you \ddot{x}^\mu =0

(This was actually my thought when making the question, if a student realises that they can bypass the difficulty of the question by doing this, it shows that they have a good understanding of covariant mechanics)

[u/suskio4 → u/Such-Shop-9724]

Already dumber than a 7th grader, the questions specified 2d roundabout so no falling!

[u/Such-Shop-9724 → u/suskio4]

yes, i guess i just assumed they were standing on a rorating disc or something like that

[u/Bendanzhang]

Bold of you to assume the children are in a vacuum and perfectly spherical.

[u/throughthehills2 → u/Bendanzhang]

Assume potato shaped children

[u/Extension_Option_122]

I mean, If you solve this you are smarter than a 7th grader, but if you fail to do so you aren't not smarter than a 7th grader.

[u/Equinoxe111]

Nothing is defined here, be more precise

[u/Digital_001 → u/Equinoxe111]

This is 7th grade physics, precision comes later

[u/TomtheMagician26]

Is this the geodesic equation?

[u/Bendanzhang]

Air resistance

[u/shartmaister]

From the ball’s point of view it doesn’t move at all

[u/AtomGutan]

7th grader Jimmy Coriolis

[u/BenjaminGal → u/AtomGutan]

A 7th grader with an interest in atmospheric sciences may actually know about the Coriolis effect, although obviously not the maths behind. lol

[u/AboveAverage1988]

They're saying 2d roundabout. Do we get to pick which two dimensions? Because I feel the problem can become significantly harder or easier depending on choice?

[u/L31N0PTR1X → u/AboveAverage1988]

I could've worded it better for sure, I intended the problem to be two dimensions in that it's top down, so gravity isn't considered. Depending on which frame you choose, you might want to use (r, 𝜃)

[u/derivative_of_life]

/uj If anyone's curious about the actual proof, this is a great lecture/lecture series: https://youtu.be/2eVWUdcI2ho

[u/L31N0PTR1X → u/derivative_of_life]

It is nice, however, to be able to introduce these concepts without something as abstract as GR, hence this nice mechanics problem that students will already be familiar with by the time they take their first GR course. Differential geometry is usually poorly introduced