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Time required by rod to fall

  1. Oct 6, 2014 #1
    1. The problem statement, all variables and given/known data
    Estimate the time it takes a pencil of length L=10 cm to hit the ground if initially the pencil makes an angle of 0.001 rad with the vertical. Take the gravitational acceleration to be 10m/s2

    Details and assumptions

    Assume that the pencil is just a thin solid rod with uniform mass distribution along its length. Also to simplify the problem, assume that the pencil's contact point with the ground (the tip) is fixed.

    2. Relevant equations


    3. The attempt at a solution

    I
    nitially the rod makes 0.001 rad with vertical.
    So it makes 89.947 deg with horizontal.
    p2.png
    T
    orque acting along point O is ##mgcos\theta \frac { l }{ 2 } ## [I will change the units of l at end]
    So

    ##\frac { dL }{ dt } =mgcos\theta \frac { l }{ 2 } ##

    or ##I\frac { d\omega }{ dt } =mgcos\theta \frac { l }{ 2 } ##

    or ##I\frac { { d }^{ 2 }\theta }{ { dt }^{ 2 } } =mgcos\theta \frac { l }{ 2 } ##

    But I don't know how to solve this second order differential equation.

    So I tried this

    ##I\omega \frac { d\omega }{ d\theta } =mgcos\theta \frac { l }{ 2 }##

    or ##I\frac { { \omega }^{ 2 } }{ 2 } =mgsin\theta \frac { l }{ 2 } +C##

    At ## t=0## ## \omega=0## and let ##\theta ={ \theta }_{ i }##

    So ## C=-mg lsin{\theta}_{i}##

    So ##I\frac { { \omega }^{ 2 } }{ 2 } =mgsin\theta \frac { l }{ 2 }-mg lsin{\theta}_{i}##

    or ##\frac { m{ l }^{ 2 } }{ 3 } \frac { { \omega }^{ 2 } }{ 2 } =mgsin\theta \frac { l }{ 2 } -mgsin{ \theta }_{ i }l##

    or ##{ \omega }^{ 2 }=\frac { 3\left\{ gsin\theta -2gsin{ \theta }_{ i } \right\} }{ l } ##

    or ##\frac { d\theta }{ dt } =\sqrt { \frac { 3\left\{ gsin\theta -2gsin{ \theta }_{ i } \right\} }{ l } } ##

    or ##\frac { \sqrt { l } }{ \sqrt { 3\left\{ gsin\theta -2gsin{ \theta }_{ i } \right\} } } d\theta =dt##

    or ##\sqrt { \frac { l }{ 3g } } \int _{ { \theta }_{ i } }^{ 0 }{ \frac { d\theta }{ \sqrt { \left\{ sin\theta -2sin{ \theta }_{ i } \right\} } } } =t##

    Am I right till here?
    How to solve this integral?
     
  2. jcsd
  3. Oct 6, 2014 #2
    You have a sign error in the first equation, which then results in negative kinetic energy and other weirdness. Remember, the usual convention is "counter-clockwise is positive".
     
  4. Oct 6, 2014 #3

    Orodruin

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    Just to add to what voko already said. Do you remember what we did in the chain-slipping-off-the-table thread? (see post #68 in that thread: Chain hanging from a table) That was a slightly more complicated differential equation, but we found a function we could multiply with to make both sides of the form ##\dot{y}(t)f(y(t))##, where ##f## is a function. Can you think of something that multiplied with the RHS of your equation makes the RHS on this form?
     
  5. Oct 6, 2014 #4
    I believe Satvik did just that, albeit with a typo in LaTeX.
     
  6. Oct 6, 2014 #5

    Orodruin

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    You are right. I guess I was confused by him first saying he did not know how to solve it and then going on to solve it ... :rolleyes:
     
  7. Oct 6, 2014 #6

    ehild

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    You made a mistake: the integration constant is C=-mgl/2 sin(thetai)

    I would name the angle enclosed with the vertical as theta. In that case, ##\omega^2 =\frac {3g}{l}\left(\cos(\theta)-\cos(\theta_i)\right)##
    The initial theta is very small and the problem text want you to estimate the time of fall. So you can approximate cos(theta)=1. (Actually, it is 0.9999995 )
    So you have ##\omega^2=\frac{3g}{l}(1-cos(\theta))##
    Here is the next trick: ##1-cos(\theta)=2 \sin^2(\theta/2)##
    Can you continue from here?

    ehild
     
  8. Oct 7, 2014 #7
    I have a confusion. Can't we assume clockwise as a positive?
    I mean does't it depends upon us whether to choose clockwise positive or anti clockwise positive?
    Oh! I missed 2 in denominator.

    If ## cos\theta_{i}## is 1
    putting this value in eq(1) yields
    ##\omega^2 =\frac {3g}{l}(cos(\theta)-1)##
    whereas your eq(2) is ##\omega^2=\frac{3g}{l}(1-cos(\theta))##
     
  9. Oct 7, 2014 #8

    ehild

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    Yes.

    I think I missed a minus sign. Check. ##\omega^2 =-\frac {3g}{l}(cos(\theta)-cos(\theta_i))##
     
  10. Oct 7, 2014 #9
    If we assume clockwise direction (-)ve then
    ##\tau =-mgsin(\theta )\frac { l }{ 2 } ##

    or ##I\omega d\omega =-mgsin(\theta )\frac { l }{ 2 } d\theta ##

    or ##I\frac { { \omega }^{ 2 } }{ 2 } =mgcos\theta \frac { l }{ 2 } +C##

    on putting the value of ##C##

    I got

    ##{ \omega }^{ 2 }=\frac { 3g }{ l } (cos\theta -1)##

    As ehild said I used this ##1-cos(\theta)=2 \sin^2(\theta/2)##


    ##{ \omega }^{ 2 }=-\frac { 3g }{ l } (1-cos\theta )##

    or ##{ \omega }=\sqrt { -\frac { 3g }{ l } 2{ sin }^{ 2 }\left( \frac { \theta }{ 2 } \right) } ##

    I got (-)ve sign inside root?:confused:

    If I choose clockwise direction as +ve then I got

    ##{ \omega }=\sqrt { \frac { 3g }{ l } 2{ sin }^{ 2 }\left( \frac { \theta }{ 2 } \right) } ##

    It looks good to me.
    So should I choose clockwise +ve?
     
  11. Oct 7, 2014 #10
    I got ##{ \omega }=\sqrt { \frac { 3g }{ l } 2{ sin }^{ 2 }\left( \frac { \theta }{ 2 } \right) } ##
    if I choose clockwise direction as +ve. Whereas in #post1 voko asked me to choose clockwise direction as -ve?
     
  12. Oct 7, 2014 #11
    One can use any convention as long as it makes sense, but the standard convention in mathematics and physics is that angles are measured counter-clockwise. If one chooses a non-standard convention, there must be a good reason for that, and I do not see any in this case.

    In the original notation, where ##\theta## was measured from the horizontal, the correct equation should be $$ I {d^2 \theta \over dt^2} = - {mgl \over 2} \cos \theta . $$

    If now the angle - I will call it ## \phi ## - is measured from the vertical, still with the convention "counter-clockwise is positive", then the equation is $$ I {d^2 \phi \over dt^2} = {mgl \over 2} \sin \phi . $$

    The second equation can be trivially obtained from the first by noticing that $$ \theta = {\pi \over 2} + \phi. $$ So there is really no major difference in using ##\phi## vs. ##\theta##, but one should stick with one choice and use it consistently (which is not the case in this thread, I am afraid).
     
  13. Oct 7, 2014 #12
    Is equation in #post 10 correct?
     
  14. Oct 8, 2014 #13

    ehild

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    It is correct, but to avoid confusion use ##\Phi## instead of ##\theta##. If you choose anti-clockwise direction positive for the angles, you need to do the same with the torque. So the torque which rotates the rod in the clockwise direction is negative.
    In your original post, theta was measured from the positive x axis in anticlockwise direction. But the torque turned the rod in clockwise direction, so it should have been taken with a minus sign. You missed that minus in the first equation. The correct equation is ## I \ddot {\theta}=-mg l/2 \cos (\theta)## as voko pointed out in Post#11.

    If you calculate with the angle from the vertical, and you stick to anticlockwise as positive direction, you can draw the falling rod to the other side of the vertical, and then both the torque and the angle are positive.
    The last advice: Always indicate the direction of the angles in your drawings.

    ehild

    fallingrod.JPG
     
  15. Oct 8, 2014 #14
    Thank you ehild and voko!

    So ##\frac { d\phi }{ dt } =\sqrt { \frac { 6g }{ l } } sin\left( \frac { \phi }{ 2 } \right) ##
    or ##\frac { d\phi }{ sin\left( \frac { \phi }{ 2 } \right) } =\sqrt { \frac { 6g }{ l } } dt##
    or ##\int _{ 0.001 }^{ \pi /2 }{ cosec\left( \frac { \phi }{ 2 } \right) } =\sqrt { \frac { 6g }{ l } } t##

    I can integrate the term in LHS.But am I right till here?
     
  16. Oct 8, 2014 #15

    ehild

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    Yes, it looks right. Integrate.
     
  17. Oct 8, 2014 #16
    ##\int _{ 0.001 }^{ \pi /2 }{ cosec\frac { \phi }{ 2 } } d\phi ##

    Let ##\frac { \phi }{ 2 } =x\quad So\quad \frac { 1 }{ 2 } =\frac { dx }{ d\phi } ##

    or ##{ d\phi }=2dx##

    or ## 2\int { cosec(x) } dx=2log{ | cosec(x)- cot(x)| }##

    or ##2log{ { | cosec\frac { \phi }{ 2 } - cot\frac { \phi }{ 2 } | } }_{ 0.001 }^{ \pi /2 }##
    Is this right?
     
  18. Oct 8, 2014 #17

    ehild

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  19. Oct 8, 2014 #18
    I got LHS ##14.82##

    So ##14.82=\sqrt { \frac { 6g }{ l } } t##
    L=0.1m

    ##t=\frac{14.82}{\sqrt{60g}} \approx 0.61sec##
    Is this right?
     
    Last edited: Oct 8, 2014
  20. Oct 8, 2014 #19

    ehild

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    I got slightly different value for the LHS, but your result is about correct.

    ehild
     
  21. Oct 8, 2014 #20
    I got this question online on the internet and answer of this question on that website is 280.71?:)
    There is no unit even!
     
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