JEE problems

This discussion is regarding some of the CM problems that Neeraj Anand Badgujar sent me; I've just started to work on some of them.

So, I was wondering how you attempted them; if you can, please provide your solution to this problem:

And this one:

Thanks; cheers! Both are pretty difficult, but I'd like to see your approach.

Note by Krishna Karthik
2 weeks, 5 days ago

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@Krishna Karthik 2nd problem(feel free to ask me anything)

Lil Doug - 2 weeks, 5 days ago

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I have a slightly different result than you:

vmax=mka0+kmx0 \bold{v}_{max} = \displaystyle \sqrt{\frac{m}{k}} a_0 + \sqrt{\frac{k}{m}} x_0

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik can you share your whole solution?

Lil Doug - 2 weeks, 5 days ago

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@Lil Doug Actually, I realised my result earlier was slightly incorrect. Now I realise the error.


So, I got the first part exactly the same as you. In my solution, xx is the position of the block relative to the left of the frame, and x0x_0 is the natural spring length.

Here's the total force as per Newton's Law:

ma=ma0k(xx0)m \bold{a} = -ma_0 - k(x-x_0)

When v\bold{v} is maximum, the derivative:

a=a0kmx+kmx0\displaystyle \bold{a} = -a_0 - \frac{k}{m}x+\frac{k}{m} x_0

is zero (same as you have done; I just took a more roundabout way to realise it lol😂).

So the value of xx which gives this is:

x=mka0+x0\displaystyle x = \frac{m}{k}a_0 + x_0

So then I equated the value of the kinetic energy to the potential; I found the potential by integrating the value of the force by the value of the stretch.

Integration:

U(s)=2ma0s+kms2\displaystyle U(s) = 2m a_0 s + \frac{k}{m} s^2

Equating the potential to the kinetic energy and substituting s=mka0\displaystyle s = \frac{m}{k} a_0, I get the same result.

I got the exact same result as you now, I just forgot the work energy theorem on the way and had to do some really complicated math lmao.

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik @Krishna Karthik nice, you were very close th answer.
BTW what is the meaning of lol??

Lil Doug - 2 weeks, 5 days ago

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@Lil Doug ?????? How come you're on the internet and you don't know the meaning of lol? It means "laughing out loud".

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik @Krishna Karthik I know that it means laughing out loud, but what is the feel of that word?

Lil Doug - 2 weeks, 5 days ago

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@Lil Doug I just used it to express the funny and roundabout nature of my method. It took me a while 'cause I forgot some rules and had to use some unnecessary calc.

Krishna Karthik - 2 weeks, 5 days ago

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@Lil Doug Hey, I have posted an explanation. In hindsight, what I did earlier was slightly incorrect; I owe that to the fact that I overcomplicated the problem by forgetting the work-energy theorem.

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik @Krishna Karthik ok lol. Try my new problem below

Lil Doug - 2 weeks, 5 days ago

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@Krishna Karthik 1st problem

Lil Doug - 2 weeks, 5 days ago

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Yes; got the exact same thing :)

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik the questions which i have shared has more difficult problems than these .

Lil Doug - 2 weeks, 5 days ago

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@Lil Doug True. The above one was easy, but the other was hard. I got both nevertheless :)

Krishna Karthik - 2 weeks, 5 days ago

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@Lil Doug I will keep working on these👍🏽

Krishna Karthik - 2 weeks, 5 days ago

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@Lil Doug Bro though, imagine doing this in an insane time-limit. That's the reason JEE's so fucking hard. This is epic.

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik @Krishna Karthik i have explained so much in my solution ,and i was doing some other work also while writing solution therefore it takes me some time to upload .
if i was in exam room it will take me just 6-8 minutes to solve both.

Lil Doug - 2 weeks, 5 days ago

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@Lil Doug Ah, true. Lol it took me 10 mins🤣 But of course, I'm not doing JEE. 6-8 mins is actually godly. I would expect smarts like that from a person trying out for JEE.

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik @Krishna Karthik While in maths ,the question are so difficult to solve and too long,
my hand starts shivering.

Lil Doug - 2 weeks, 5 days ago

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@Lil Doug Same. I just can't give a fuck to read long questions or shit. I usually just skip them and try to get the more straightforward questions first. But in Australia, maths is a joke. Like actually. If people do normal Australian math, you'd be doing grade 7 (in India) at grade 9 (in Australia). That's why I switched to IB lol.

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik @Krishna Karthik jee physics feels me easy ,but in maths i always fuck up.

Lil Doug - 2 weeks, 5 days ago

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@Lil Doug Yeah, well, maths in JEE's always super hard. Like, who even needs to solve integrals in their head? Seriously. A lot of the questions are about sums, series, and other really complicated shit that requires some kind of trick to figure out.

Krishna Karthik - 2 weeks, 5 days ago

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@Lil Doug Tbh if I'm doing JEE math I would fuck up. I'm not very "math competition material". In math competitions sometimes I figure out the trick, other times, I just don't. And I'm not good with time. I hate problems that have some dumb trick or something. Just so pointless. That's why I like to use math for physics.

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik @Krishna Karthik well if one has done practice a lot ,then one can solve those problems .

Lil Doug - 2 weeks, 5 days ago

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@Krishna Karthik try this one

Lil Doug - 2 weeks, 5 days ago

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I feel it will be a parabola. If you have attempted it mathematically, that's super brilliant. Pls tell me how!

But my logic is this:

The distance from the CM to the radial surface of it (to the bottom of the hemisphere) is less, so the CM will gradually move down and then up as the hemisphere rolls. I think that's why a parabola can be the only way it moves. It can't move in a cyclic manner, because it doesn't roll fully. And a straight line isn't an option, because we can see it will move down and up.

Edit: bruh

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik The answer is straight line.
Because after when string will burnt,
No force is acting in horizontal direction.
And the force mgmg is acting in centre of mass in downward direction.
Therefore trajectory will be straight line

Lil Doug - 2 weeks, 5 days ago

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@Lil Doug Ahhh, ok. Thanks mate! I originally thought straight line, but stupidly switched to parabola lol.

Krishna Karthik - 2 weeks, 5 days ago

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@Krishna Karthik @Krishna Karthik BTW
The word “Parabola” feels me little stylish , therefore people often chose it . Ha ha

NJ STAR - 2 weeks, 5 days ago

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@Nj Star Ha yea lol. Idk; Somehow, I knew that the CM will move in a straight line (maybe I read it in a book somewhere). Kinda funny.

Krishna Karthik - 2 weeks, 5 days ago

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@Neeraj Anand Badgujar @NJ STAR

What did you get for problem 41? Pls check out the note that I just made :)

Krishna Karthik - 2 weeks, 3 days ago

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