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Rotational Mechanics

  1. Dec 25, 2008 #1
    A rod standing on frictionless table(Rotation)

    1. The problem statement, all variables and given/known data
    A rod of length l is standing on a friction less surface.
    A slight impulse is given to the rod and hence the rod starts falling. Find the torque,angular acceleration ,angular velocity, the normal force by the ground and the distance through which the end part of the rod slides on the ground when the rod makes an angle of [tex]\theta[/tex] withe the vertical


    2. Relevant equations

    Law of conservation of energy and equations of moment of forces.

    3. The attempt at a solution

    We can apply the laws of conservation of energy

    [tex] mgh = mg\frac{l}{2}(1-cos\theta) + something [/tex]

    I dont know what this something would be. It may be the sum of the rotational energy and something. It cant just be rotational energy cause the rod is also sliding on the surface.

    From this we would get angular velocity. Differentiating it would give us angular acceleration. Now we can apply the equation of moments. Let x be the distance by which the rod slides.

    [tex]N\frac{l}{2}cos\theta = I\alpha [/tex]

    The above is about the centra of mass.

    [tex]Mg\frac{l}2}sin\theta = I_{0}\beta[/tex]

    The above is about the point which touches the ground(ie the endpoint)

    What we have to do is to find a relation between alpha and beta and the acc which we get from the law of conservation of energy.

    Moreover these are just three equations while we have four variables ( alpha, beta, N and x) and we have no equation containing x.
     
    Last edited: Dec 25, 2008
  2. jcsd
  3. Dec 25, 2008 #2
    In the first and second equation the h and the x should not be there. I am trying to edit but vain
     
  4. Dec 25, 2008 #3

    Doc Al

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    Staff: Mentor

    Re: A rod standing on frictionless table(Rotation)

    The total KE of the rod.

    Hint: What path does the center of mass take?
     
  5. Dec 25, 2008 #4
    Re: A rod standing on frictionless table(Rotation)

    Does total mean rotational + translational. If yes than what would be the relation between linear velocity and angular velocity. I dont think that it would be
    [tex] v = \frac{l}{2}\omega[/tex]

    The path should be a parabola. But still that is not helpful. Lets suppose that initially the co ordinate of the endpoint of the rod is (0,0) then the position of the centre of the mass of the rod once it becomes horizontal would be less than l/2 cause the rod would have slided a certain distance in the -ve x direction.

    though i can apply

    [tex] \frac{l}{2} - x = vt[/tex]

    [tex]\frac{l}{2} = \frac{1}{2}gt^{2}[/tex]

    Hence

    [tex]\sqrt{\frac{g}{l}}(\frac{l}{2} - x) = v_{x}[/tex]

    And what else can i apply?

    Please help.

    We have obtained the above equation but we have also added a variable (ie V_x)

    We definetly assume V_x to be constant

    Now at [tex]\theta[/tex]

    [tex]v_y = \sqrt{2gh}[/tex]

    [tex] h = \frac{l}{2} - xtan\theta - \frac{l}{2}cos\theta + x(cosec\theta)(cos\theta) [/tex]

    This is weird.
     
  6. Dec 25, 2008 #5

    Doc Al

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    Staff: Mentor

    Re: A rod standing on frictionless table(Rotation)

    Yes. Translational KE of the center of mass plus rotational KE about the center of mass.
    Use a bit of geometry. Find the position of the rod's center as a function of angle.

    No, it's simpler than that. Hint: Consider the forces acting on the rod.
     
  7. Dec 26, 2008 #6
    Then there is just one way left. A straight line. And in that case then the center of mass finally would be at (0,0).

    I can definetly apply geometry once i know the point about which i have to take the velocities. I have done various problems but not the ones in which the point about which the object is rotating is sliding.

    Just elaborate a few important points eg how to determine x at a certain angle, the relation between [tex] \alpha & \beta[/tex]
     
  8. Dec 27, 2008 #7

    Doc Al

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    Staff: Mentor

    Right. The center of mass falls straight down since there is no horizontal force on the rod.
    What's the relationship between the position of the center of mass and the angle of the rod?
     
  9. Dec 27, 2008 #8
    [tex] y_{cm} = \frac{l}{2}cos\theta[/tex]

    Yup thats better. Now a final relation between alpha and beta is needed. But as you have said that the path is a straight line i feel that alpha and beta are the same.

    I must say thing. The way you help people is ingenious. Thats because you help in such a way that one feels thayt he or she has done the problem on his own. Cause you do not give direct solutions but you give deep insights. Its like " Dont offer a fish to the hungry, but teach him how to fish"
     
  10. Dec 28, 2008 #9
    Hey Doc

    You have really helped me in this problem. And i am about to finish it only if you could just comment on my last post.
     
  11. Dec 28, 2008 #10

    Doc Al

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    Staff: Mentor

    Good.
    What do you mean by "alpha" and "beta"?

    Thank you. There is a method to my madness. :wink:
     
  12. Dec 28, 2008 #11

    Doc Al

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    Staff: Mentor

    Re: A rod standing on frictionless table(Rotation)

    OK, I see how you use alpha and beta. They are the same.
    This is OK.

    This relationship is not necessarily true. The point of contact accelerates.
     
  13. Dec 28, 2008 #12
    The point of contact accelerates. Lets say instead of beta we use

    [tex]
    Mg\frac{l}2}sin\theta = I_{0}\sqrt{{\beta}^2 + ({\alpha}\frac{l}{2})^2 - 2{\beta}{\alpha}{\frac{l}{2}}cos{\theta}} [/tex]
     
  14. Dec 28, 2008 #13

    Doc Al

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    Staff: Mentor

    Re: A rod standing on frictionless table(Rotation)

    That should be sinθ, not cosθ.
    Why not just forget about that point? The torque about the center of mass is the one you need.
     
  15. Dec 29, 2008 #14
    Bingo.

    Forget the end point. In the first post i have written 3 equations forget the last one.

    Now instead of something we can use

    [tex] \frac{1}{2}I(\omega)^2 + \frac{1}{2}m(2g)(cos\theta)\frac{l}{2} [/tex]

    And we will diff it so that we will get an equation in alpha.

    Now we have got 3 variables and 3 equations. Sum solved.

    Thank you.
     
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