problem 43

So, here's problem 43:

It asks for the acceleration of the bodies' centre of mass when they are a distance of D/2D/2 apart.

Here's a more rigourous mathematical approach:

The centre of mass can be found using xˉ=x1+2x23\displaystyle \bar{x} = \frac{x_1 + 2x_2}{3}; with this, we can find the second derivative of xˉ\bar{x} with respect to time, giving:

xˉ¨=13x¨1+23x¨2\displaystyle \ddot{\bar{x}} = \frac{1}{3} \ddot{x}_1 + \frac{2}{3} \ddot{x}_2

So, now we compute the gravitational force which can give the values of x¨1\ddot{x}_1 and x¨2\ddot{x}_2:

Fg=2GM2r2\displaystyle \bold{F}_g = \frac{2GM^2}{r^2}

x¨1=2GMr2\displaystyle \ddot{x}_1 = \frac{2GM}{r^2}

x¨2=GMr2\displaystyle \ddot{x}_2 = -\frac{GM}{r^2}

So, acceleration of C.M is:

xˉ¨=13x¨1+23x¨2\displaystyle \ddot{\bar{x}} = \frac{1}{3} \ddot{x}_1 + \frac{2}{3} \ddot{x}_2

xˉ¨=132GMr223GMr2\displaystyle \ddot{\bar{x}} = \frac{1}{3} \frac{2GM}{r^2} - \frac{2}{3} \frac{GM}{r^2}

Substituting r=D/2r = D/2, xˉ¨=138GMD2234GMD2\displaystyle \ddot{\bar{x}} = \frac{1}{3} \frac{8GM}{D^2} - \frac{2}{3} \frac{4GM}{D^2}

xˉ¨=8GM3D28GM3D2\boxed{\displaystyle \ddot{\bar{x}} = \frac{8GM}{3D^2} - \frac{8GM}{3D^2}}

Therefore the acceleration, xˉ¨=0\boxed{\displaystyle \ddot{\bar{x}} = 0}

Note by Krishna Karthik
4 months ago

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And also go to this link

Talulah Riley - 4 months ago

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Ah, now I have the intuitive understanding of what's going on. The result is not very intuitive to understand in my opinion. But the problem is still there that my mathematics contradicts what actually happens. Can you spot the error in the maths?

Krishna Karthik - 4 months ago

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@Krishna Karthik The net external force acting on the system is zero. Therefore position,velocity,acceleration of center of mass will be constant.
Answer is D

Talulah Riley - 4 months ago

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@Krishna Karthik you have missed r2r^{2} in those 3 expression

Talulah Riley - 4 months ago

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OH my GAWWWD I'm so dumb🤣🤣 Mistake of the century👍🏽👍🏽👍🏽

Krishna Karthik - 4 months ago

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@Krishna Karthik you have still not corrected it.

Talulah Riley - 4 months ago

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@Talulah Riley Going to, soon.

Krishna Karthik - 4 months ago

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@Talulah Riley Gimme a sec; writing the latex down

Krishna Karthik - 4 months ago

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@Talulah Riley Alright, it's done. Corrected. Looking nice :)

Krishna Karthik - 4 months ago

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I also forgot that the direction of the x¨2\ddot{x}_2 acceleration is negative. That makes more sense. It starts to cancel out as well.

Krishna Karthik - 4 months ago

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You can prove it by displacing a particle xx and one according to force and write down their displacement from origin.
And then again calculating centre of mass.
You will notice that centre of mass is not changing its position , so if it is not changing its position then definitely it has 0 velocity and 0 acceleration.
I don't think we have to use pen/pencil in this problem.
This problem can be solved by our feelings.
When you started feeling physics,the need of applying mind will be less.

Talulah Riley - 4 months ago

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Yes, true. Thank you for telling me m8. Self-learning physics is so much fun for me :)

Krishna Karthik - 4 months ago

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@Krishna Karthik Hii bro.
I am not able to solve a integration using euler explicit method
Will you help me??

Talulah Riley - 4 months ago

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@Neeraj Anand Badgujar Integration, or ODE solving?

Krishna Karthik - 4 months ago

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Integration

NJ STAR - 4 months ago

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@Nj Star Ok, I'll make a note soon.

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik No bro.
I will give you integral and you will give me code of that particular integral.
Nothing else I want.

NJ STAR - 4 months ago

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@Nj Star Alright, got it.

Krishna Karthik - 4 months ago

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@Nj Star Cool. :)

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik you doing need to make code

NJ STAR - 4 months ago

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@Nj Star Ok; where's the integral?

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik No matter how much time it will take.
I want answer, and if you give me code , I will also run it.
So we can match our answer

NJ STAR - 4 months ago

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@Nj Star The code seems fine, but what the problem is is that it is so computationally expensive, I'm not even joking lmao.

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik what does computationally expensive mean??

NJ STAR - 4 months ago

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@Nj Star What I mean is that when you run crude and brute code like mine on the CPU, it'll take ages to compute. When I say ages, I mean ages.

I'll try to come up with a better piece of code in the meanwhile, and reduce the amount of cycles so it is compilable, but it'll take a little bit of time. You don't mind waiting, do you?

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik what does ages mean???
Give the code to me, I will run it in python and will leave the computer for whole 1 day.

NJ STAR - 4 months ago

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@Nj Star Lmao🤣Nah fam, there are better ways of computing your integral. I'll try to find a nice numerical method (not simple brute-force) which I can implement.

Krishna Karthik - 4 months ago

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@Nj Star By the way, do you need only the code? Have you computed the integral yourself, or not? And, lastly, are you using the integral for any special purpose?

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik no I have not calculated the integral.
I was making a problem, so this integration has come.
Now please provide me code.
I also want to run.

NJ STAR - 4 months ago

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@Nj Star Right now, for your problem, to get a decent value of the integral, I've had to use 3 while-loops each with a 1000 different iterations. Even if I reduce it to a hundred, the iteration count would be something like 1003100^3.

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik are you using python??

NJ STAR - 4 months ago

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@Nj Star Yeah.

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik share the code with me,
I will also run.

NJ STAR - 4 months ago

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@Nj Star Hold up, I don't even know if it can work perfectly because it takes ages to even produce a result. Lol. Just wait a few minutes or something; I'll get a better piece of code.

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik what does ages mean??

NJ STAR - 4 months ago

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@Nj Star Have you tried using Monte-Carlo method for calculating the integral?

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik I didn't know monte Carlo.

NJ STAR - 4 months ago

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@Nj Star That's what I'm saying, give me some time to do this. I'll do it bro. :)

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik ok bro.
But i just want to see, what code is?

Talulah Riley - 4 months ago

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@Talulah Riley Ok, I'll give you the code. I'll post it in a discussion.

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik the integral is little bit big, read it carefully.

NJ STAR - 4 months ago

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@Nj Star Got it. I will need to code from scratch a triple integral engine. Hold up.

Krishna Karthik - 4 months ago

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@Nj Star I need to ask a question; what is the variable on the very right of the denominator being squared? What is that expression?

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik (sinysinz)2(\sin y \sin z)^{2}

NJ STAR - 4 months ago

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@Nj Star Ah. Thanks. The "s" from sin(z)\sin(z) confused me a bit lol

Krishna Karthik - 4 months ago

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@Nj Star I can't understand what the expression on the very right of the denominator says.

Krishna Karthik - 4 months ago

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@Krishna Karthik got it or not??

NJ STAR - 4 months ago

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Yeah yeah, coming. Coding from scratch takes a while lol. Gimme about 30 mins.

Krishna Karthik - 4 months ago

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@Krishna Karthik which tool you are using bro?
Python?? Midpoint? Explicit??

NJ STAR - 4 months ago

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@Nj Star python. For this kinda thing it will take a while. I have to redesign the code for triple-integral. I'll be done fairly soon. I'll let you know when I'm done.

Krishna Karthik - 4 months ago

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@Nj Star Cheers bro!

Krishna Karthik - 4 months ago

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@Neeraj Anand Badgujar

How have you posted the problem when the integral is incorrect?

Krishna Karthik - 4 months ago

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@Lil Doug

Help me pls; I'm working on a new problem and kinda need people to verify the physics.

Krishna Karthik - 3 months, 1 week ago

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@Krishna Karthik bro i need your help.

Talulah Riley - 2 months ago

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With what? btw it's 12 here. I'm gonna go to sleep soon, so tell me quickly.

Krishna Karthik - 2 months ago

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@Krishna Karthik it is not urgent.
I have very much doubts in mechanics, while posting in brilliant, people clear my doubts very slowly.
Is there any website where I will get Pathfinder book solution ??

Talulah Riley - 2 months ago

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@Talulah Riley I don't know of any specific thing. Look on youtube; there'll always be people doing solutions for the harder problems.

Krishna Karthik - 2 months ago

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@Krishna Karthik please don't mind, I want to say something
If you keep asking this very simple doubts, I am going to completely ignore you
And don't dare to think that I am copying someone. 😂😂😂😂

Talulah Riley - 4 months ago

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Bruh is that a joke or u being serious right now

Krishna Karthik - 4 months ago

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@Krishna Karthik I am joking.
Try to recall this lines where said by my favourite physics teacher.

Talulah Riley - 4 months ago

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@Talulah Riley Ah yeah, I remember lol. Yeah; he got pretty pissed.

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik he is still my favourite

Talulah Riley - 4 months ago

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@Talulah Riley Of course, of course. Yeah, well, he does still help a lot. And there aren't very many engineers, physicists, or mathematicians would willingly and patiently help the younger generation and mould them into successful people.

Krishna Karthik - 4 months ago

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@Krishna Karthik @Krishna Karthik Yeah Exactly. He is a genius and a very intelligent man.

By the way, if you have more doubts on my questions, you can ask now.
Otherwise I have to give you some more hard problems.

Talulah Riley - 4 months ago

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@Talulah Riley All good man, when I do the rest of problems, let's see.
Thanks mate! These are very interesting problems.

Get this: In physics (at school, grade 10) we are doing displacement-time graphs.

And here's the definition of vector that we have to parrot:

"A vector is an object with magnitude and direction" XD

We haven't even gotten to kinematic stuff yet🤣🤣

Krishna Karthik - 4 months ago

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