# problem 43

So, here's problem 43:

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

Here's a more rigourous mathematical approach:

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

$\displaystyle \ddot{\bar{x}} = \frac{1}{3} \ddot{x}_1 + \frac{2}{3} \ddot{x}_2$

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

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

$\displaystyle \ddot{x}_1 = \frac{2GM}{r^2}$

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

So, acceleration of C.M is:

$\displaystyle \ddot{\bar{x}} = \frac{1}{3} \ddot{x}_1 + \frac{2}{3} \ddot{x}_2$

$\displaystyle \ddot{\bar{x}} = \frac{1}{3} \frac{2GM}{r^2} - \frac{2}{3} \frac{GM}{r^2}$

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

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

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

Note by Krishna Karthik
7 months, 1 week ago

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And also go to this link

- 7 months, 1 week ago

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?

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

@Krishna Karthik you have still not corrected it.

- 7 months, 1 week ago

Going to, soon.

- 7 months, 1 week ago

Gimme a sec; writing the latex down

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

You can prove it by displacing a particle $x$ 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.

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

@Neeraj Anand Badgujar Integration, or ODE solving?

- 7 months, 1 week ago

Integration

- 7 months, 1 week ago

Ok, I'll make a note soon.

- 7 months, 1 week ago

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

- 7 months, 1 week ago

Alright, got it.

- 7 months, 1 week ago

Cool. :)

- 7 months, 1 week ago

@Krishna Karthik you doing need to make code

- 7 months, 1 week ago

Ok; where's the integral?

- 7 months, 1 week ago

@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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

@Krishna Karthik what does computationally expensive mean??

- 7 months, 1 week ago

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?

- 7 months, 1 week ago

@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.

- 7 months, 1 week ago

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.

- 7 months, 1 week ago

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?

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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 $100^3$.

- 7 months, 1 week ago

@Krishna Karthik are you using python??

- 7 months, 1 week ago

Yeah.

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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.

- 7 months, 1 week ago

@Krishna Karthik what does ages mean??

- 7 months, 1 week ago

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

- 7 months, 1 week ago

@Krishna Karthik I didn't know monte Carlo.

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

@Krishna Karthik got it or not??

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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.

- 7 months, 1 week ago

Cheers bro!

- 7 months, 1 week ago

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

- 7 months, 1 week ago

@Lil Doug

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

- 6 months, 2 weeks ago

@Krishna Karthik bro i need your help.

- 5 months, 2 weeks ago

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

- 5 months, 2 weeks ago

@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 ??

- 5 months, 2 weeks ago

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

- 5 months, 2 weeks ago

@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. 😂😂😂😂

- 7 months, 1 week ago

Bruh is that a joke or u being serious right now

- 7 months, 1 week ago

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

- 7 months, 1 week ago

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

- 7 months, 1 week ago

@Krishna Karthik he is still my favourite

- 7 months, 1 week ago

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.

- 7 months, 1 week ago

@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.

- 7 months, 1 week ago

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🤣🤣

- 7 months, 1 week ago