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!

#Mechanics

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Comments

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 years, 1 month ago

This seems to be a generalization of that "underkill" problem that was posted recently.

Steven Chase - 4 years, 1 month 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

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

Brian Charlesworth - 4 years, 1 month ago

@Brian Charlesworth – It would be interesting to consider this same problem with a curved surface and friction.

Steven Chase - 4 years, 1 month ago

@Steven Chase – Okay! So go on, Can you post one?

Md Zuhair - 4 years, 1 month ago

@Md Zuhair – I'll look into it

Steven Chase - 4 years, 1 month ago

Which underkill problem? Can you please provide the link sir?

Md Zuhair - 4 years, 1 month ago

@Md Zuhair – https://brilliant.org/problems/mechanics-overkill-2/?ref_id=1349100

Steven Chase - 4 years, 1 month ago

@Steven Chase – Thank you,

Md Zuhair - 4 years, 1 month 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 $t_{1}$

Rohith M.Athreya - 4 years, 1 month ago

@Rohith M.Athreya – Hey! Whats your JEE Main score?

Md Zuhair - 4 years, 1 month ago

@Md Zuhair – around 290 :(

Rohith M.Athreya - 4 years, 1 month ago

@Rohith M.Athreya – Well, lets see your picture in newspaper!

Md Zuhair - 4 years, 1 month ago

@Md Zuhair – why?not at all!

my friends are expecting around 310-315

Rohith M.Athreya - 4 years, 1 month ago

@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 years, 1 month ago

@Md Zuhair – well....

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

thank you!

Rohith M.Athreya - 4 years, 1 month ago

@Rohith M.Athreya – If you dont write advanced, ill delete my Brilliant account!

Md Zuhair - 4 years, 1 month ago

@Rohith M.Athreya – So @Rohith M.Athreya are you giving either of ISI or CMI ?

Aditya Narayan Sharma - 4 years, 1 month ago

@Aditya Narayan Sharma – CMI yes

II no

Rohith M.Athreya - 4 years, 1 month ago

Thanks sir!

Md Zuhair - 4 years, 1 month ago

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

@Md Zuhair From where did you get this problem?

Ankit Kumar Jain - 4 years, 1 month ago

i did a very fine mistake! Its correct now!

Md Zuhair - 4 years, 1 month ago

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

Ayon Ghosh - 4 years, 1 month ago

Ya I got it now

Md Zuhair - 4 years, 1 month ago

Let me tag some of you,

@Aniket Sanghi , @Rohith M.Athreya , @Ayon Ghosh , @Steven Chase , @Brian Charlesworth!

Md Zuhair - 4 years, 2 months ago

http://olympiads.hbcse.tifr.res.in/olympiads/wp-content/uploads/2017/01/INPhO2017-Solution-20170131.pdf

Rohith M.Athreya - 4 years, 1 month 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}`	$\boxed{123}$