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

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 :-( . · 4 months ago

Ya I got it now · 4 months ago

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. · 4 months ago

This seems to be a generalization of that "underkill" problem that was posted recently. · 4 months ago

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

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

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

Okay! So go on, Can you post one? · 4 months ago

I'll look into it · 4 months ago

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 $$t_{1}$$ · 4 months ago

Hey! Whats your JEE Main score? · 4 months ago

around 290 :( · 4 months ago

Well, lets see your picture in newspaper! · 4 months ago

why?not at all!

my friends are expecting around 310-315 · 4 months ago

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

well....

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

thank you! · 4 months ago

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

CMI yes

II no · 4 months ago

If you dont write advanced, ill delete my Brilliant account! · 4 months ago

Which underkill problem? Can you please provide the link sir? · 4 months ago

https://brilliant.org/problems/mechanics-overkill-2/?ref_id=1349100 · 4 months ago

Thank you, · 4 months ago

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

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

i did a very fine mistake! Its correct now! · 4 months ago

Thanks sir! · 4 months ago

Let me tag some of you,

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