Looks are sceptical!
A disc of radius R is spun to an angular speed ω about its axis and then imparted a horizontal velocity of magnitude ω R / 4 at t = 0 with its plane remaining vertical.
The coefficient of friction between the disc and the plane is μ . The sense of rotation is anticlockwise and linear speed is in forward direction. Find the time taken by the disc to return to its initial point.
Details and Assumptions:
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If your answer is T . Then define f = μ g T / ω R , where g is the acceleration due to Gravity on earth.
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If f = a / b for positive co-prime integers a , b .
Find the value of a + b .
The answer is 73.
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2 solutions
Yeah !!!! please explain
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Have you drawn the graph ...... at x>0 y>0 the slope is continuously decreasing so you will get at exactly one normal here......for x<0 .y<0..if you draw lines from 0,1 in this region lines would have a positive slope and graph too would have a positive slope here so no chance of normal
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i got the second case why x<0 and y<0 no normal would be there . product of normal and graph has to be -1 . but did not got why exactly one normal in x>0 and y>0
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@Prakhar Bindal – That you can easily imagine by seeing the graph ,if not then try by joining points on graph with the given point ,you will see that........Mathematically , the slope of lines from 0,1 drawn in this region continuously decreases in magnitude (angle from x-axis increases) while the slope of tangents on the graph continuously decreases (angle from x-axis decreases) hence after one point of normal their can't be any other on increasing x after that
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@Aniket Sanghi – I Think i got it! thanks . if i have any other confusion i will surely ask
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@Prakhar Bindal – You are (most) welcome ... Brother! :)
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@Aniket Sanghi – i found that for x>0 , y>0 as the slope of both are decreasing there can be only one normal through (0,1) not more than that. But can we say that normal has to exist in this interval? . Is it possible that no normal to the curve exists in x>0 , y> 0 . How can we prove this
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@Prakhar Bindal – Basically how can we prove that a normal has to exist in this interval ? @Aniket Sanghi
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@Prakhar Bindal – On careful observation we find that the angle between tangents at a point and the line joining from 0,1 to that point increases from 0 to 180 degrees
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@Aniket Sanghi – Yeah thats true
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@Prakhar Bindal – Did you got any idea?
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@Prakhar Bindal – About what?
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@Aniket Sanghi – about explaining this fact
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@Prakhar Bindal – i am talking of the proof that atleast one normal will exist to the curve
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@Prakhar Bindal – Angle continuously increases from 0 to 180 , so exactly one 90
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@Aniket Sanghi – So you mean to say that exactly one normal will have one angle 90 . that is vertical normal?.
this implies that x axis must be the tangent to the curve right?
but dy/dx for the curve = 9/4y^2 . Hence y axis will be tangent to the curve passing through (0,1) not normal
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@Prakhar Bindal – Oh no , I was talking about angle between the line from given point and the tangent to curve
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@Aniket Sanghi – Yeah that is correct angle will increase. how does that prove that atleast one normal will exist for x>0 and y>0
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@Prakhar Bindal – Since it is increasing continuously, therefore
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@Aniket Sanghi – oh got it!! That was an awesome explanation! Thanks a lot.
I Was underestimating the question it was nice to do it graphically(completely taking each and every case into account)
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@Prakhar Bindal – Try to do as many q as possible , graphically , it will save your time
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@Aniket Sanghi – Most of my friends did it graphically but did not analyzed each and every case just jumped to a conclusion which yielded them correct answer but was techically wrong as they did not thought about other possbilities
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@Prakhar Bindal – So did my solution included each and every case
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@Aniket Sanghi – Yeah now your solution is complete and includes each and every case.
Thanks bro!
@Prakhar Bindal – Hmm give me time to derive an explanation by that time you can try my new Chem q
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@Aniket Sanghi – Ohhk!! i will try my best ( i am too bad at chemistry i feel these days due to aldehyde and ketone)
@Aniket Sanghi – I Think i need to revise cyclohexane as i have forgotten almost everything taught to us In GOC
@Aniket Sanghi – You should try this (Its one of the best question i encountered till now in calculus)
https://brilliant.org/problems/how-will-this-be-done/?ref_id=1177541
@Prakhar Bindal – The angle between the lines increases continuously so there can be only one 90 degree
this is hard!
just view for the point when v=wr it comes after t=5wr/12ug and find the displacement till now. next since now velocity is constant(pure rolling) and in opposite to till displacement vector note time more required to travel it and hence calculate total=25wr/48ug
Yeah but it is a bit calculative and lengthier than it looks in solution
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Could you please post a solution for this question if disc is in horizontal plane?
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It would Be a nice exercise. i will analyze and report. by the way how was mains?
yes if you have a shorter method then please share and i don,t wanted to write the whole solution as it would be :p
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Nahh! It came in our workshop worth 6 marks in a subjective paper of 80!
Hey what are you being taught in fiitjee currently? when is the expected date of syllabi completion? what has been taught to you of class 12 till now?
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@Prakhar Bindal – we are still Snar (not yet started) capacitors(just started) function (even and odd)
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@Aryan Goyat – Ok. Weren't you taught 12th topics in 11th? still you are left with the torture of aldehyde and ketone!
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@Prakhar Bindal – no we were not and whats your progress.
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@Aryan Goyat – We were taught a lot of 12th topics in 11th itself like Matrices and determinants , Alkyl and aryl halide, Alcohol,Ether And Phenol , Aldehyde and ketone (except tests for aldehyde and ketone) , Geometrical Optics. Currently We are on Application of derivatives in maths, Practical Organic chemistry and biomolecules , And Current electricity (about to start)
@Aryan Goyat – As our sir told us in chemistry we are only left with 3 chapters of physical and 4 chapters of inorganic. hence we might end syllabi in 3 months.
Physics will go on till september as we are with normal batches.
Maths we are left with 4 chapters in calculus, Probability,Vector 3d. so again we might end maths in 3 months
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@Prakhar Bindal – so you would have a lot of time to revise and test.
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@Aryan Goyat – Yeah might be! . But i have got a lot of backlogs like Fluid mechanics, Alcohol ether phenol , Ionic equilibrium , P&C etc
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@Prakhar Bindal – ionic and P&C are damm too challenging and tricky.
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@Aryan Goyat – Also i really struggle in surface tension and capillary rise it goes over my head
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@Prakhar Bindal – they can be easy just do 10 challenging questions of each and you will be the best in it for sure!!!!!
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@Aryan Goyat – Thanks. i will give them extra time during summer vacations . Bye ! my mother is shouting over my head to sleep and if i dont listen to her i may get scolded
@Aryan Goyat – i think in october syllabus may complete.
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@Aryan Goyat – Seriously the most dreaded chapter i faced in chemistry after p block is aldehyde and ketone! . The JEE Archive has over 325 problems! . It has hell lot of about 35 named reaxn's. i dont remember even 5 :P
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@Prakhar Bindal – i have a little bit different problem that i don't give even 50% of what i am in exams may be i get nervous i spend a lot of time struggling with calculations and so on and never able to complete paper at time require about 1hr more.
@Prakhar Bindal – may be i too learn to stay cool during exams.
@Aryan Goyat – We have lot of time (over an year to overcome these) ! :)
I can't talk you there that page is hanging ... are you online @Prakhar Bindal