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Physics Olympiad Problem

Two identical blocks A and B each of mass M are placed on a long inclined plane (angle of inclination = θ ) with A higher up than B. The coefficients of friction between the plane and the blocks A and B are respectively µ A and µ B with tan θ > µ B > µ A. The two blocks are initially held fixed at a distance d apart. At t = 0 the two blocks are released from rest.

Find the time in which the two blocks collide?

My answer ; - \[\large{t_{Collision} = \sqrt{\dfrac{2d \tan \theta ( \sin \theta - \mu_A \cos\theta)}{g (\sin \theta - \mu_B \cos\theta)}}}\]

What about you?Is my answer correct?

I am not sure with my answer, please all of us, lets discuss!

Note by Md Zuhair
4 months ago

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@Md Zuhair Yup, my answer matches with the one given by Brian Charlesworth and yes, its correct :-) ! Sorry yours was wrong :-( . Ayon Ghosh · 4 months ago

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@Ayon Ghosh Ya I got it now Md Zuhair · 4 months ago

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The two blocks will meet up after a time \(t\) such that

\(d + \dfrac{g}{2}(\sin(\theta) - \mu_{B}\cos(\theta)) = \dfrac{g}{2}(\sin(\theta) - \mu_{A}\cos(\theta))\),

which when solved for \(t\) gives us \(t_{c} = \sqrt{\dfrac{2d}{g\cos(\theta)(\mu_{B} - \mu_{A})}}\).

Suppose \(\mu_{A} = \mu_{B}\). Then you wouldn't expect the two blocks to ever meet, but your formula would have them meeting after \(\sqrt{\dfrac{2d\tan(\theta)}{g}}\) seconds, while mine would have \(t_{C} \to \infty\) as expected. Brian Charlesworth · 4 months ago

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@Brian Charlesworth This seems to be a generalization of that "underkill" problem that was posted recently. Steven Chase · 4 months ago

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@Steven Chase Exactly. I found it a bit odd at first that \(\cos(\theta)\) was in the denominator, since this would result in a finite \(t_{c}\) for \(\theta = 0\), which doesn't make sense, (although an infinite \(t_{c}\) for \(\theta = \pi/2\) does make sense as the two blocks would both be in free-fall so A would never catch up). But then I remembered that we require that

\(\tan(\theta) \gt \mu_{B} \Longrightarrow \theta \gt \arccos\left(\dfrac{1}{\sqrt{1 + \mu_{B}^{2}}}\right)\). Brian Charlesworth · 4 months ago

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@Brian Charlesworth It would be interesting to consider this same problem with a curved surface and friction. Steven Chase · 4 months ago

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@Steven Chase Okay! So go on, Can you post one? Md Zuhair · 4 months ago

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@Md Zuhair I'll look into it Steven Chase · 4 months ago

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@Steven Chase One can generalise further : If the time at which they first meet is \(t_{1} \), the subsequent times of meeting again (after collision) turns out to be odd multiples of \(t_{1} \) Rohith M.Athreya · 4 months ago

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@Rohith M.Athreya Hey! Whats your JEE Main score? Md Zuhair · 4 months ago

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@Md Zuhair around 290 :( Rohith M.Athreya · 4 months ago

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@Rohith M.Athreya Well, lets see your picture in newspaper! Md Zuhair · 4 months ago

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@Md Zuhair why?not at all!

my friends are expecting around 310-315 Rohith M.Athreya · 4 months ago

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@Rohith M.Athreya Okay, leave that! You will surely qualify JEE Advanced with rank above 0.5k this year brother! I have seen your brain! Md Zuhair · 4 months ago

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@Md Zuhair well....

I'll take your word for it(if i write advanced)

thank you! Rohith M.Athreya · 4 months ago

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@Rohith M.Athreya So @Rohith M.Athreya are you giving either of ISI or CMI ? Aditya Narayan Sharma · 4 months ago

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@Aditya Narayan Sharma CMI yes

II no Rohith M.Athreya · 4 months ago

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@Rohith M.Athreya If you dont write advanced, ill delete my Brilliant account! Md Zuhair · 4 months ago

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@Steven Chase Which underkill problem? Can you please provide the link sir? Md Zuhair · 4 months ago

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@Md Zuhair https://brilliant.org/problems/mechanics-overkill-2/?ref_id=1349100 Steven Chase · 4 months ago

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@Steven Chase Thank you, Md Zuhair · 4 months ago

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@Brian Charlesworth Even I got the same answer as you sir. :)

@Md Zuhair From where did you get this problem? Ankit Kumar Jain · 4 months ago

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@Ankit Kumar Jain i did a very fine mistake! Its correct now! Md Zuhair · 4 months ago

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@Brian Charlesworth Thanks sir! Md Zuhair · 4 months ago

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Let me tag some of you,

@Aniket Sanghi , @Rohith M.Athreya , @Ayon Ghosh , @Steven Chase , @Brian Charlesworth! Md Zuhair · 4 months ago

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@Md Zuhair http://olympiads.hbcse.tifr.res.in/olympiads/wp-content/uploads/2017/01/INPhO2017-Solution-20170131.pdf Rohith M.Athreya · 4 months ago

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