Problem no 41

3 gravitating bodies; find the work done by gravity.

What answer did you get for this question? And if you don't mind, can you pls share your method?

I'm getting $\displaystyle W = -\frac{3GM^2}{2l}$ ;

I integrated the force by the displacement; this results in this complicated integral:

$\large \displaystyle \int_{0}^{\frac{2}{\sqrt{3}} l} \frac{\sqrt{3}gm^2}{ \left(2l - \frac{2}{\sqrt{3}}r \right)^2 } dr$

That's 'cause I totally forgot about energy-work theorem lol

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Can't we do that work in macbook.?
Is it necessary to purchase ipad or it is just a waste of money.

Talulah Riley - 4 weeks ago

@Talulah Riley iPads are generally a lot cheaper than Macbooks. A good mac costs 1700 AUD or above, and the best iPad only costs around 1100 AUD. Tablets have their own uses. One is that you can write on them as if they are a notebook with an Apple pencil. You can do mobile gaming like PUBG extremely well on a tablet.

However, iPads are certainly not a waste of money. They are my favourite products from Apple and have their own unique features.

Krishna Karthik - 4 weeks ago

I don't want gaming as I said earlier ,but i am planning to buy ipad air .
Cost of ipad pro in India:72k Indian Ruppee.
Cost of ipad air in India:45k Indian Ruppee.
I can easily buy ipad air ,but not sure for pro.
My Father used to say ,only buy those thing which you need and are important, don't just buy to show off.

Talulah Riley - 4 weeks ago

@Talulah Riley – Precisely. I don't understand why you would buy a pro in India; it's much much more than the iPad air. In Australia, the pro costs only 100 dollars more than the iPad Air, so it's very different. I would only consider getting the pro in Australia.

Pricing is so very different in India.

Krishna Karthik - 4 weeks ago

@Krishna Karthik what do you think about ipad air ?
In my opinion iPad pro is very costly,so it is good to but ipad air.
It costs around 0.5 lakh ruppee which is very less.

Talulah Riley - 4 weeks ago

My sister just recently got an iPad pro for 1100 AUD; it's really good. I'm not sure about the prices in India however. I think the iPad Air is value for money in India. Around 1000 dollars seems reasonable for the latest iPad Air.

Krishna Karthik - 4 weeks ago

@Krishna Karthik tell me difference between apple ipad and macbook

Talulah Riley - 4 weeks ago

@Talulah Riley – An iPad is a very easy-to-use and useful piece of tech. Macbooks are just regular computers, iPhones are just smartphones, but I would describe the iPad as something in between. Tablets have their own features and uniqueness. You can use a tablet for gaming, note-taking, design, drawing, working, and even other stuff.

Krishna Karthik - 4 weeks ago

@Krishna Karthik but after 5-6 years I will definitely be ultra rich.so I will buy the latest best apple phone,at that time.

Talulah Riley - 4 weeks ago

Haha. Time will tell :) Perhaps you can run a business like Musk. Since you're specialising in maths and physics in high school, it makes sense that you make money buy starting a business. I'm taking business in high school too, maybe I can try to create a business of my own when I move out of university.

Krishna Karthik - 4 weeks ago

@Krishna Karthik Yeah bro.i have started studying electrical engineering.
From 2025 My electric cars will be in the market .more efficient and cheaper than Tesla

Talulah Riley - 4 weeks ago

@Talulah Riley – Good. Great to think big; you have to get into IIT, and then maybe get into MIT.

Krishna Karthik - 4 weeks ago

@Krishna Karthik – @Krishna Karthik Yeah . don't worry everything will be fine.i am doing best from my side.
Let's see what happens.

Talulah Riley - 4 weeks ago

@Krishna Karthik Bye ,I am going for lunch.will talk with you later ,when I will be free.bye .

Talulah Riley - 4 weeks ago

@Krishna Karthik I am going to buy products from Dubai.
Due to this I will have the opportunity for tourism also.

Talulah Riley - 4 weeks ago

Right, see you later. Have a nice day :)

Krishna Karthik - 4 weeks ago

Nice. When you have the chance to go to Dubai, make sure to get an iPhone 12 there.

Krishna Karthik - 4 weeks ago

@Krishna Karthik currently I have Android of around 20k Indian Rupee.
But after 4 months I am going to buy iphone 11.

Talulah Riley - 4 weeks ago

@Talulah Riley Are you an apple fan? Personally, I like Android more than apple for my phones.

Krishna Karthik - 4 weeks ago

@Krishna Karthik Yeah ,in India if you have an apple product ,people think you are rich.
Nothing else.
But in America ,the tax system is not that hard as Indian.
Therefore Iphones are very common in America.

Talulah Riley - 4 weeks ago

@Talulah Riley – iPhones are pretty common in Australia too. All my classmates have iPhones. Frankly, around here if you have an iPhone, you'll just look really normal and basic. I got an Android because I wanted some variety. Switched out of an iPhone 6. And I liked Samsung better.

Krishna Karthik - 4 weeks ago

@Krishna Karthik – @Krishna Karthik Anyone have iphone 12 pro max?
I don't think so.

Talulah Riley - 4 weeks ago

@Talulah Riley – Yeah, a few people have the iPhone 12 pros, and most have iPhone 11s. I think some of the kids that take film have an iPhone 12 Pro Max because of the amazing camera.

Krishna Karthik - 4 weeks ago

@Krishna Karthik – @Krishna Karthik I want to buy iphone 12 pro max 128 GB gold .
But it around 1.3 lakh ruppee in India.My father will not allow for that.
In my opinion instead of buying such expensive phone I should invest money in laptop.

Talulah Riley - 4 weeks ago

@Talulah Riley – Of course. 1.3 lakhs is a total waste of money on a phone; I wouldn't allow it either. Next time when you go to a foreign country, get an iPhone there. A laptop is a much better way to invest your money like that.

Krishna Karthik - 4 weeks ago

Talulah Riley - 4 weeks ago

@Talulah Riley Lmfao. This is so funny. I never understood why Apple prices their phones so high in India, no wonder the smartphone market in India is dominated by Android.

What phone do you have by the way?

Krishna Karthik - 4 weeks ago

@Krishna Karthik By the way ,in India specifically Iphones are very very costly.

Talulah Riley - 4 weeks ago

@Talulah Riley Yeah, Apple is generally very costly in India, which is why people in India tend to use Android more. I just choose Android as a matter of preference, I don't like Apple's phones that much.

Krishna Karthik - 4 weeks ago

@Krishna Karthik Iphones are very costly.but apple laptops not that much .

Talulah Riley - 4 weeks ago

@Talulah Riley – Yeah. Macs look about the same price in India as they do in Australia. It's crazy how much iPhone 12 pro max costs: 2100 AUD in India! Mental. Some iPhones in India cost over 1 lakh rupees. I can never understand why a phone would cost that much.

I recently bought a Samsung Galaxy S20 FE (this is the latest budget flagship from Samsung) for only 800 AUD, which is 0.44 lakhs. In India phones happen to be much more expensive, and I don't know for what reason...

Krishna Karthik - 4 weeks ago

@Krishna Karthik Just tell me is this a good desktop for me ?

Talulah Riley - 4 weeks ago

@Krishna Karthik that iMacs contains GPU?
Are Desktop type laptops more expensive than general apple laptop. ? Because ,I like the design of Desktops.

Talulah Riley - 4 weeks ago

Yeah; looks good. If you go on the Indian website it is under your budget, and has SSD, 8GB RAM (8GB seems good enough), and an i5 processor. You can overclock your CPU to 3.6 Ghz so that's nice, if you ever want that extra boom.

Krishna Karthik - 4 weeks ago

@Talulah Riley Desktops aren't necessarily more expensive than laptops of the same specs. In fact, laptops tend to be more expensive when they have the same specs as desktops, because laptop technology includes the display, the compact technology, and extra features.

If you like desktops, you have many options available.

Getting a gaming desktop is one option. In my opinion, it's a good option because you are getting amazing hardware for low prices. However, I see that you like the Mac OS and the video editing on it.

In that case, you can get an iMac without a GPU and try to get a good CPU and a good amount of RAM. RAM is important in video editing and programming and so is a good CPU.

Krishna Karthik - 4 weeks ago

@Krishna Karthik Does apple laptop contains GPU?

Talulah Riley - 4 weeks ago

Actually, I found a few iMacs that have GPUs: https://www.apple.com/au/shop/buy-mac/imac

They have radeon graphics, which is good dedicated graphics. It's good. And some of it is within your price range.

If you want good video editing, go for it. Buy an iMac with 16GB RAM and a GPU (If you can find one in your price range).

Krishna Karthik - 4 weeks ago

@Krishna Karthik is GPU really necessary in a laptop for beginner and for video editor ?

Talulah Riley - 4 weeks ago

@Talulah Riley But if you want to get a beginner laptop, try to get a mac. Make sure it has 16 GB RAM, at least a core i7 processor, and an Intel UHD integrated graphics chip. It's up to you; if you want, you can exclude the GPU and get a mac in instead.

Krishna Karthik - 4 weeks ago

@Krishna Karthik share the link of laptop of apple?

Talulah Riley - 4 weeks ago

@Talulah Riley – Actually, I'd rather you have an iMac because it is better hardware and value for money: https://www.apple.com/au/shop/buy-mac/imac

Krishna Karthik - 4 weeks ago

@Krishna Karthik – @Krishna Karthik All 3 of them doesn't contain GPU :)

Talulah Riley - 4 weeks ago

@Talulah Riley – Yeah they do. Radeon is a graphics chip. However, they all have only Intel i3 processors and only about 8 GB RAM. This is what I mean about you having to pay a lot more for a good mac. If you really really want a mac, for your price range, you will have to get it without a GPU.

Krishna Karthik - 4 weeks ago

@Krishna Karthik – @Krishna Karthik Bro the 1st laptop out of that 3 laptips is a good one or not ?

Talulah Riley - 4 weeks ago

@Talulah Riley – Yeah, it looks like it has good specs. And it's under your budget, so that's nice. Intel Iris graphics is nice, despite the fact that it isn't dedicated graphics. Looks good :)

Krishna Karthik - 4 weeks ago

@Krishna Karthik – @Krishna Karthik so which laptop has best dedicated graphics.
My condition is that I will take apple only.

Talulah Riley - 4 weeks ago

@Talulah Riley – @Talulah Riley Sorry for the late response, I was momentarily a bit busy. Out of the desktops I just showed you, desktops 2 and 3 have the same dedicated graphics, except desktop 3 has twice as much video memory.

The Macbook pro laptops have Apple graphics, but I don't know much about apple graphics. Out of the laptops, I think it's pretty hard to get a laptop within your price range that has a good GPU. I'd say get a desktop.

Krishna Karthik - 4 weeks ago

@Talulah Riley Often, with a price range up till 1 lakh, a GPU is necessary. It wouldn't be very smart to buy a 1 lakh laptop that doesn't have a dedicated GPU. I would recommend getting an Nvidia GPU.

1 lakh, there is no reason not to get a GPU. However, if you don't want one, that's totally fine. But a GPU is pretty useful.

If you want your laptop only for video editing, I would consider getting a mac. However, a Windows laptop is ok for video editing most of the time. But make sure to get 16 GB RAM! RAM is very important for video editing. I think at your price range, you should get a GPU too. In the long run you'll find it more useful.

Krishna Karthik - 4 weeks ago

@Steven Chase @NJ STAR

I need help with a new problem I'm posting; it's an error in Matlab code (coded in python style so it's easy to understand for python users too). I've posted it in a new note.

Krishna Karthik - 6 months, 1 week ago

@NJ STAR @Steven Chase @Neeraj Anand Badgujar

Hey guys; I kinda need help with a problem (problem 41). I've posted it in a new note. If you can help me that would be awesome. Cheers!

Krishna Karthik - 7 months ago

@Krishna Karthik Hii sir. Try this problem

Talulah Riley - 7 months ago

Will do it soon! :)

Krishna Karthik - 7 months ago

@Krishna Karthik would you like to play chess now?

Talulah Riley - 7 months ago

@Talulah Riley – Ok, let's play 10 min game

Krishna Karthik - 7 months ago

@Talulah Riley – Oh lol I'm at school, but we're doing some dumb enrichment thing. I have no problem playing chess :)

Krishna Karthik - 7 months ago

@Krishna Karthik it is not work done by gravity. It is workdone by external agent..

Talulah Riley - 7 months ago

@Neeraj Anand Badgujar @NJ STAR Ah, just for your interest, here is the integral which I calculated to solve the problem:

$\large \displaystyle \int_{0}^{\frac{2}{\sqrt{3}} l} \frac{\sqrt{3}gm^2}{ \left(2l - \frac{2}{\sqrt{3}}r \right)^2 } dr$

Krishna Karthik - 7 months ago

Which is gravity, basically.

Krishna Karthik - 7 months ago

@Krishna Karthik it can be other different forces also.

Talulah Riley - 7 months ago

@Talulah Riley – Yeah, but why would there be a gravitational constant in the equation otherwise? Or, is it possible that a mechanical force would need the exact same amount of work to move the balls inward by half?

Krishna Karthik - 7 months ago

@Krishna Karthik – @Krishna Karthik yes it is possible. If the balls come up with their own interacting force then they all have some velocity.

Talulah Riley - 7 months ago

@Talulah Riley – Oh ok. Thanks for explaining. Bro you are expert in physics.

Krishna Karthik - 7 months ago

I just woke up.

Talulah Riley - 7 months ago

@Talulah Riley I just got a notification that you commented on this discussion. Did you delete your comment?

Krishna Karthik - 4 weeks, 1 day ago

@Krishna Karthik yes I commented to that guy Rohan Joshi .
But he deleted his comment,which eventually lead to delete of my comment.
He was telling you the way ,how to solve the problem.
And I was just commented in a very sarcastic way,so he deleted his comment.
Nothing else :).
By the way bro ,how are you ?,I am fine. I just want to ask you which laptop is best for programming and coding for heavy loading things. Suggest me.

Talulah Riley - 4 weeks ago

@Talulah Riley – I'm good bro. I hope you're fine too since India's schools are still shut down and all that.

I'd be happy to give you a suggestion... usually a laptop with a good CPU and a good amount of RAM will be able to run most code (Matlab, python, OpenGL etc.) well. The minimum I would say to be a laptop with an Intel Core i7 CPU and at least 16 GB of RAM at a speed of around 2666 Mhz.

Sometimes, if you want to speed up your code the extra mile, you may need to use a GPU. However, if you're just getting started with coding, you may not need to use a GPU, since programming for the GPU is more complicated than just regular coding, like we've been doing with Python and Matlab.

GPUs are very helpful in rendering graphics. General purpose computing at a really high level for scientists and physicists often uses CUDA and all that. You can choose whether you want a GPU or not, but the only main things you need is an SSD, a good amount of RAM, and a decent CPU.

Gaming laptops are often a good way to get great specs for a good value for money. I recommend Lenovo, HP, and MSI if you want to get a cheap gaming laptop.

But, I warn you. Most cheap gaming laptops start at around 1300 AUD. I don't know what they are priced at in India however. But converting to INR, it's around 0.7 lakh rupees. It's a worthy investment.

Krishna Karthik - 4 weeks ago

@Krishna Karthik – @Krishna Karthik I don't want a gaming laptop.
My budget is around 1.2 lakh rupees. I was thinking to buy this one
I don't know why but I am impressed by Apple desktop that is iMac..
Is there any difference in buying laptop and desktop of apple.?

Talulah Riley - 4 weeks ago

@Talulah Riley – In my opinion, Mac is generally more expensive than most computers. Also, if you want to get good specs with a mac, you need to pay a lot more. For 1.2 lakhs, you can get a computer with awesome specs and can certainly afford a GPU.

Don't use 1.2 lakhs on a mac with less RAM.

In my experience, one of the only good ways to get a computer with a good graphics card and amazing specs without spending too much is to get a gaming laptop. Now, this doesn't mean the laptop is for gaming.

Many professionals and developers spend their money on a gaming laptop because they don't want to spend an extra 1000 dollars on a professional GPU. Most people can't even afford a professional GPU. Most of the time, gaming GPUs offer much more features for their price.

I have a gaming computer and most of the time I don't even use it for gaming! I mostly use it for GPU computation, OpenGL, and Matlab. It really really helps, trust me.

Look here: Are gaming PCs good for programming?

For 2000 AUD (around 1 lakh in INR) you should easily be able to find a PC with a good GPU. Go ahead and get a GPU with it, if you don't, you're missing out on something that's so helpful.

Krishna Karthik - 4 weeks ago

@Krishna Karthik – @Krishna Karthik bro i have to edit a lot of videos .
And in internet I heard that apple laptop are the best to edit videos.

Talulah Riley - 4 weeks ago

@Talulah Riley – You could purchase from HP. HP is pretty nice. If you want Apple, and if you like the software, consider getting a Macbook Air, or a 13 inch pro. However, they do exceed your budget. And you have to pay a lot to get it with a GPU. If you want to do video editing with a mac and a budget of 1.2 lakhs, you'll have to get it without a GPU, because otherwise it seriously exceeds your budget.

Krishna Karthik - 4 weeks ago

@Talulah Riley – Edit videos? What for? I thought you wanted it for programming. Nevertheless, the Mac OS is good for video editing. Personally, I have a Mac laptop from 2015 and a good graphics laptop.

However, video editing can be done well on Windows too. The main thing is however is that when you buy a mac I feel like you are spending way too much money just to get a machine that doesn't have as much ram and lacks a GPU. You could be using the exact same amount of money to get a laptop also with a GPU and 16 GB RAM.

The GPU is quite important and it's not good to miss out on one. By the way, video rendering is much faster if you have a GPU. My sister has done a lot of video editing on her mac. Now, the software was great, but it took her 2 hours to render something and her computer was getting really hot.

This was because she didn't have a GPU. On a GPU, it would have merely taken minutes to render stuff.

Krishna Karthik - 4 weeks ago

@Talulah Riley – @Krishna Karthik.
I dont want gaming.
I want a good high level programming computer.
I want a good editing laptop.

Talulah Riley - 4 weeks ago

@Talulah Riley – You don't seem to understand... many high end laptops are just called "gaming laptops". This doesn't mean they're for gaming. In the slightest. They just have good hardware.

Gaming laptops are not always for gaming. For many, they are just quick ways to get great hardware. Professionals from various industries use "gaming" laptops.

Krishna Karthik - 4 weeks ago

Actually, this can be solved using process of elimination as well; firstly, we know that the work done must be negative, and for that there are only two options.

Secondly, we know that the $M$ term of the gravitational force must be of order two ( $M^2$ ), therefore the only suitable option for the work must be (A).

Krishna Karthik - 7 months ago

Very good! Btw I solved the problem using a really complicated integral lol. I integrated the force by the displacement along the path of the objects, and thereby got the final solution.

Krishna Karthik - 7 months ago

@Krishna Karthik very nice.
The formula which I have used comes from your method only. . I just have used shortcut.

Talulah Riley - 7 months ago

@Talulah Riley – Yeah; I really need to familiarise myself with work energy theorem lol.

Krishna Karthik - 7 months ago

I just ran a numerical time-domain simulation and looked at the total kinetic energy at the end, assuming that no external force is trying to slow it down. And then the answer is just the negative of that, which turns out to be the first option (A).

import math

# Constants

M = 2.0
L = math.sqrt(3.0)/2.0
G = 1.0

dt = 10.0**(-5.0)

###################################

# Initialize simulation

t = 0.0

y = 0.0

x = 1.0    # start on x axis at 1 distance unit away
xd = 0.0
xdd = 0.0

# Run simulation

while x >= 0.5:

    x = x + xd*dt       # numerical integration
    xd = xd + xdd*dt

    x2 = -0.5*x         # location of other particle
    y2 = math.sqrt(3.0)*x/2.0

    Dx = x2 - x
    Dy = y2 - y

    D = math.hypot(Dx,Dy)

    ux = Dx/D
    uy = Dy/D

    F = 2.0*G*M*M/(D**2.0)  # double the force for two particles

    Fx = F*ux  # net force only in horizontal direction

    xdd = Fx/M

    t = t + dt

###################################

E = 3.0*0.5*M*(xd**2.0)   # total kinetic energy of three particles


opt1 = -3.0*G*(M**2.0)/(2.0*L)  # compare against options
opt2 = 3.0*G*(M**2.0)/(2.0*L)
opt3 = G*(M**2.0)/L
opt4 = -3.0*G*M/(2.0*L)

print dt
print E
print ""
print opt1
print opt2
print opt3
print opt4

#>>> 
#1e-05
#6.92824539557

#-6.92820323028
#6.92820323028
#4.61880215352
#-3.46410161514
#>>>

Steven Chase - 7 months ago

Thanks! Yes; turns out I did get the correct answer. Thanks for this beautiful solution :) Computers are always brilliant.

Krishna Karthik - 7 months ago

@Krishna Karthik whenever he uses the variable of problem as constant.
Like in this problem he has used $M=2.0$ it made me laugh 😂
He is a very smart guy😎.

Talulah Riley - 7 months ago

@Talulah Riley – yea lol; but literally any variable you plug into values of $M , G, L$ it will give you same thing. Programmers like to be efficient😎

Krishna Karthik - 7 months ago

@Krishna Karthik – @Krishna Karthik yeah I used this smart tricks in maths in my exam.
The way he chooses random values make me laugh.

Talulah Riley - 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$