Falling and colliding Pencil!

Classical Mechanics Level 5

See both the angles are same......This may not be possible but assume it... (to make calculations easy)....also rod is translating.(assumption)

A pencil is placed vertically with its tip on the edge of a table. It is then given a slight push so that it starts falling towards the ground. It makes collision with ground such that its velocity in vertical direction reduces to zero.

What is its angular velocity just after collision?

* Details and Assumptions *

  • Length of pencil l = 0.6 m l = 0.6\text{ m} , the height of table is h = 3888 78125 h = \dfrac{3888}{78125} , Mass per unit length of pencil varies as λ = x \lambda = x ,(where, x x is the distance from the tip of the pencil), take g = 10 m/s 2 g = 10\text{ m/s}^2 .

  • (Important note) Assume that when the pencil detaches from table surface It performs pure translational motion.Neglect any change in its com velocity due to this.that is just after it detaches ,assume ω = 0 \omega = 0 ( just to make it simpler).

  • There is sufficient friction on the table. There is sufficient friction on the ground such that just after collision there is no relative motion between the contact surfaces.

  • Better to solve the question by taking the quantities given in fraction form , as fraction only.

  • Neglect air drag.

Try more here


The answer is 5.00.

Aniket Sanghi by Aniket Sanghi , 21, India

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2 solutions

Aniket Sanghi
Apr 16, 2016

Centre of mass distance from tip = 2l/3 ; moment of inertia of pencil about com is m l 2 18 \frac{ml^2}{18} ; moment of inertia about tip m l 2 2 \frac{ml^2}{2}

Check when will the normal reaction becomes zero... you will get c o s θ = 16 25 cos\theta = \frac{16}{25}

Velocity at the time of losing contact

Horizontal velocity =128/125(Vx)

Vertical velocity = 72/125

Velocity at the time of collision ; vertical = 144/125(Vy). Horizontal is the same.

Vertical component of velocity reduces to zero(as given)

Ndt = MVy

Similarly , f dt = M (Vx - V)

And the angular impulse equation is

N d t ( 2 l s i n θ / 3 ) + f d t ( 2 l c o s θ / 3 ) = m l 2 ω 18 Ndt (2lsin\theta/3) + f dt(2lcos\theta/3) = \frac{ml^2\omega}{18}

Since friction will stop the relative motion therefore velocity of bottom most point

V = ω ( 2 l c o s θ / 3 ) \omega(2lcos\theta/3)

Now on solving we get ω = 5.00 \omega = \boxed{5.00}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

This question would probably cost you more than 30 minutes.so rough calculations.what I think is that you should retain the originality of the problem let the angles be different.when already we have so much to so what's wrong in doing extra.good question.

Spandan Senapati - 4 years, 8 months ago

Sorry.I did nt want to question on the authenticity of the question.I praise your invincing originality.I would like you to help me sort out if I did anything wrong.well I conserved the angular momentum about the point of contact.the problem after the first essentially leads to one of your similar problems.but why do you very often prefer to use the impulse momentum theorem.is it because of the fact that there can be some non inertial frames as well and we do need to add up their velocities .

Spandan Senapati - 4 years, 8 months ago

Yah most if your questions are good.I like em all.you are in fiitjee raipur by the way.I m in fiitjee Bhubaneswar.

Spandan Senapati - 4 years, 8 months ago

No.I am in class 11.we have had only 1 each of monthly test for cm batch,phase1(jee main-rank 4,adv-4) and aiits(52 air).one of my friends has got air1 in aiits 1..aiits 2 on 17 oct

Spandan Senapati - 4 years, 8 months ago

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Good ! As I recall , my AIITS Rank was 8th Your Phase test and AIITS Rank is Good , keep it up! 👍

Aniket Sanghi - 4 years, 8 months ago

Hey! I'm getting Vx and cos(theta) at the time contact is lost the same as yours but my vy is not matching. Can you please help me out? @Aniket Sanghi

Thomas Jacob - 3 years, 3 months ago

this question took me 10 mins to do and 1 failed attempt nice question ! it is almost astonishing that the answer came integer !

1 Helpful 0 Interesting 0 Brilliant 0 Confused

I create the question using variables and at last supply values so that the answer comes in simple terms , either in integer or some good rational number! ! That's why here the answer is in integer!

Even that critical value of height is because that will give simple answer !!!

Aniket Sanghi - 4 years, 6 months ago

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you have good sense of creating questions !

A Former Brilliant Member - 4 years, 6 months ago

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