1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Falling rod, finding velocity of CM

  1. Oct 22, 2013 #1
    1. The problem statement, all variables and given/known data
    A rod of mass m and length l is held vertically on a smooth horizontal floor. If the rod begins to fall from this position, find the speed of its CM when it makes an angle ##\theta## with vertical.


    2. Relevant equations



    3. The attempt at a solution
    This is a solved problem in my book and it is solved using the energy method. I was wondering if this can be solved through Newton's laws.

    The forces acting on the rod are its own weight and normal reaction from the floor.
    From Newton second law:
    $$mg-N=ma \,\,\,\, (*)$$

    Taking torque about the CM:
    $$N\frac{l}{2}\sin\theta=\frac{ml^2}{12}\alpha \,\,\,\, (**)$$
    where ##\alpha## is the angular acceleration about CM. Since ##\alpha=a/(l/2)##, I substitute this in (**) and solve for N. From here I get a relation between N and a. Substituting this relation in (*), I get:
    $$a=\frac{3g\sin\theta}{1+3\sin\theta}$$
    Since, ##a=v(dv/dy)## and ##y=l/2(1-\cos\theta)##,
    $$a=2v\frac{dv}{l\sin\theta d\theta}$$

    Therefore,
    $$2vdv=\frac{3gl\sin^2\theta}{1+3\sin\theta}d\theta$$
    The next step is to integrate both the sides. I could not integrate the right hand side so I used Wolfram Alpha.
    http://www.wolframalpha.com/input/?t=crmtb01&f=ob&i=integrate sin^2(x)/(1+3sin(x)) dx

    Wolfram Alpha gives a very strange answer and it does not match with the book's final answer.

    Are my equations wrong? :confused:

    Any help is appreciated. Thanks!
     

    Attached Files:

  2. jcsd
  3. Oct 22, 2013 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi Pranav-Arora! :smile:
    nooo :redface:

    (btw, your use of τ = Iα is valid, since this is a 1D case, and d/dt rc.o.m x vc.o.m = 0)
     
  4. Oct 22, 2013 #3
    Hi tiny-tim! :)

    Why? I don't see any error there. :confused:
     
  5. Oct 22, 2013 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    a is the (vertical) acceleration of the centre of mass

    a = rα is correct when the point at distance r (in this case, the end of the rod) is fixed (and then the centre of mass is moving in a circle)

    here, the end of the rod is moving, the centre of mass is not moving in a circle, y = l/2 cosθ, and a = y'' :smile:
     
  6. Oct 22, 2013 #5
    I am not sure if I get it. How do I find N then? Since ##\alpha=a/(l/2)## is invalid, do I have to substitute ##\alpha=d^2\theta/dt^2##?
     
  7. Oct 22, 2013 #6

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

  8. Oct 22, 2013 #7
    Solving for N, I get:
    $$N=\frac{ml}{6\sin\theta}\frac{d^2\theta}{dt^2}$$
    Substituting in (*)
    $$mg-\frac{ml}{6\sin\theta}\frac{d^2\theta}{dt^2}=m\frac{d^2y}{dt^2} \,\,\,\, (***)$$

    Since ##y=(l/2)(1-\cos\theta)##, ##dy/dt=(l/2)\sin\theta d\theta/dt##, therefore,
    $$\frac{d^2y}{dt^2}=\frac{l}{2}\sin\theta \frac{d^2\theta}{dt^2}+\frac{l}{2}\cos\theta\frac{d\theta}{dt}$$
    Substituting the above in (***) gives a second order differential equation but solving this gives ##\theta## as a function of time and I need the velocity as a function of ##\theta##. :confused:
     
  9. Oct 22, 2013 #8

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    (your dθ/dt should be squared)

    you can eliminate t by putting ω = dθ/dt,

    and d2θ/dt2 = dω/dt = dω/dθ dθ/dt = ω dω/dθ :smile:
     
  10. Oct 22, 2013 #9
    But I need velocity as a function of angle, not the angular velocity. :confused:
     
  11. Oct 22, 2013 #10

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    i didn't say it was easy! :smile:

    this is why it's so much easier to use energy :wink:
     
  12. Oct 22, 2013 #11
    Yes, you are right. I recognised the complexity when it lead me to a second order differential equation.

    But still, is there no way to find velocity from angular velocity?
     
  13. Oct 22, 2013 #12

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    you can solve that differential equation for ω as a function of θ,

    and then use v = l/2 ω sinθ :wink:
     
  14. Oct 22, 2013 #13
    Thanks a lot tiny-tim! :smile:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Falling rod, finding velocity of CM
Loading...